The interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics
aa r X i v : . [ m a t h . A C ] J u l The interplay of Invariant Theory withMultiplicative Ideal Theory and with ArithmeticCombinatorics
K´alm´an Cziszter and M´aty´as Domokos and Alfred Geroldinger
Dedicated to Franz Halter-Koch on the occasion of his 70th birthday
Abstract
This paper surveys and develops links between polynomial invariants of fi-nite groups, factorization theory of Krull domains, and product-one sequences overfinite groups. The goal is to gain a better understanding of the multiplicative idealtheory of invariant rings, and connections between the Noether number and the Dav-enport constants of finite groups.
Key words: invariant rings, Krull monoids, Noether number, Davenport constant,zero-sum sequences, product-one sequences
The goal of this paper is to deepen the links between the areas in the title. Invarianttheory is concerned with the study of group actions on algebras, and in the presentarticle we entirely concentrate on actions of finite groups on polynomial algebrasvia linear substitution of the variables.To begin with, let us briefly sketch the already existing links between the men-tioned areas. For a finite-dimensional vector space V over a field F and a finitegroup G ≤ GL ( V ) , let F [ V ] G ⊂ F [ V ] denote the ring of invariants. Since E. Noetherwe know that F [ V ] G ⊂ F [ V ] is an integral ring extension and that F [ V ] G is a finitelygenerated F -algebra. In particular, F [ V ] G is an integrally closed noetherian domainand hence a Krull domain. Benson [4] and Nakajima [58] determined its class group.Krull domains (their ideal theory and their class groups) are a central topic in mul- MTA Alfr´ed R´enyi Institute of Mathematics, Re´altanoda u. 13 – 15, 1053 Budapest, Hungary,
Email addresses: [email protected], [email protected] ; Institute for Math-ematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz,Austria,
Email address: [email protected] tiplicative ideal theory (see the monographs [46, 51] and the recent survey [52]). B.Schmid [73] observed that the Noether number of a finite abelian group G equals theDavenport constant of G (a constant of central importance in zero-sum theory) andthis established a first link between invariant theory and arithmetic combinatorics.Moreover, ideal and factorization theory of Krull domains are most closely linkedwith zero-sum theory via transfer homomorphisms (see [40, 37] and Subsection3.2).These links serve as our starting point. It is well-known that a domain R is aKrull domain if and only if its monoid R • of nonzero elements is a Krull monoid ifand only if R (resp. R • ) has a divisor theory. To start with Krull monoids, a monoid H is Krull if and only if its associated reduced monoid H / H × is Krull, and everyKrull monoid H is a direct product H × × H where H is isomorphic to H / H × . Areduced Krull monoid is uniquely determined (up to isomorphism) by its character-istic (roughly speaking by its class group C ( H ) and the distribution of the primedivisors in its classes; see the end of Subsection 4.2). By definition of the classgroup, a Krull monoid H is factorial if and only if C ( H ) is trivial. Information onthe subset C ( H ) ∗ ⊂ C ( H ) of classes containing prime divisors is the crucial ingre-dient to understand the arithmetic of H , and hence in order to study the arithmetic ofKrull monoids the first and most important issue is to determine C ( H ) ∗ . By far thebest understood setting in factorization theory are Krull monoids with finite classgroups where every class contains a prime divisor. Indeed, there has been an abun-dance of work on them and we refer the reader to the survey by W.A. Schmid in thisproceedings [77]. A canonical method to obtain information on C ( H ) ∗ is to identifyexplicitly a divisor theory for H . A divisor theory of a monoid (or a domain) H is adivisibility preserving homomorphism from H to a free abelian monoid which satis-fies a certain minimality property (Subsection 2.1). The concept of a divisor theorystems from algebraic number theory and it has found far-reaching generalizations inmultiplicative ideal theory ([51]). Indeed, divisor-theoretic tools, together with ideal-theoretic and valuation-theoretic ones, constitute a highly developed machinery forthe structural description of monoids and domains.All the above mentioned concepts and problems from multiplicative ideal theoryare studied for the ring of invariants. Theorem 4.5 (in Subsection 4.2) provides anexplicit divisor theory of the ring of invariants R = F [ V ] G . The divisibility preservinghomomorphism from R • goes into a free abelian monoid which can be naturallydescribed in the language of invariant theory, and the associated canonical transferhomomorphism q : R • → B ( C ( R ) ∗ ) from the multiplicative monoid of the ring R onto the monoid of zero-sum sequences over the class group of R also has a naturalinvariant theoretic interpretation. In addition to recovering the result of Benson andNakajima on the class group C ( F [ V ] G ) (our treatment is essentially self-contained),we gain further information on the multiplicative structure of R , and we pose theproblem to determine its characteristic (Problem 1). In particular, whenever we canshow – for a given ring of invariants – that every class contains at least one primedivisor, then all results of factorization theory (obtained for Krull monoids withfinite class group and prime divisors in all classes) apply to the ring of invariants. nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 3 In Subsection 4.3 we specialize to abelian groups whose order is not divisibleby the characteristic of F . The Noether number b ( G ) is the supremum over all fi-nite dimensional G -modules V of the maximal degree of an element in a minimalhomogeneous generating system of F [ V ] G , and the Davenport constant D ( G ) is themaximal length of a minimal zero-sum sequence over G . We start with a result on thestructural connection between F [ V ] G and the monoid of zero-sum sequences over G ,that lies behind the equality b ( G ) = D ( G ) . Clearly, the idea here is well known (asfar as we know, it was first used by B. Schmid [73], see also [24]). The benefit of thedetailed presentation as given in Proposition 4.7 is twofold. First, the past 20 yearshave seen great progress in zero-sum theory (see Subsection 3.4 for a sample ofresults) and Proposition 4.7 allows to carry over all results on the structure of (long)minimal zero-sum sequences to the structure of G -invariant monomials. Second, weobserve that the submonoid M G of R • consisting of the invariant monomials is againa Krull monoid, and restricting the transfer homomorphism q : R • → B ( C ( R ) ∗ ) (mentioned in the above paragraph) to M G we obtain essentially the canonical trans-fer homomorphism M G → B ( C ( M G ) ∗ ) . This turns out to be rather close to thetransfer homomorphism y : M G → B ( b G ) into the monoid of zero-sum sequencesover the character group of G (see Proposition 4.7), which is responsible for theequality b ( G ) = D ( G ) . The precise statement is given in Proposition 4.9, which ex-plains how the transfer homomorphism y (existing only for abelian groups) relatesto the more general transfer homomorphism q from the above paragraph which ex-ists for an arbitrary finite group. In Proposition 4.9 we point out that every class of C ( F [ V ] G ) contains a prime divisor which contributes to Problem 1.Let now G be a finite non-abelian group. Until recently the precise value of theNoether number b ( G ) was known only for the dihedral groups and very few smallgroups (such as A ) . In the last couple of years the first two authors have determinedthe precise value of the Noether number for groups having a cyclic subgroup ofindex two and for non-abelian groups of order 3 p [13, 10, 12]. In this work resultson zero-sum sequences over finite abelian groups (for example, information on thestructure of long minimal zero-sum sequences and on the k th Davenport constants)were successfully applied. Moreover, a decisive step was the introduction of the k th Noether numbers, a concept inspired by the k th Davenport constants of abeliangroups. The significance of this concept is that it furnishes some reduction lemmas(listed in Subsection 5.1 ) by which the ordinary Noether number of a group can bebounded via structural reduction in the group.The concept of the k th Davenport constants D k ( G ) has been introduced by Halter-Koch [50] for abelian groups in order to study the asymptotic behavior of arithmeti-cal counting functions in rings of integers of algebraic number fields (see [40, The-orem 9.1.8], [67, Theorem 1]). They have been further studied in [15, 30]. In thelast years the third author and Grynkiewicz [39, 48] studied the (small and the large)Davenport constant of non-abelian groups, and among others determined their pre-cise values for groups having a cyclic subgroup of index two. It can be observed thatfor these groups the Noether number is between the small and the large Davenportconstant. K´alm´an Cziszter and M´aty´as Domokos and Alfred Geroldinger
This motivated a new and more abstract view at the Davenport constants, namely k th Davenport constants of BF-monoids (Subsection 2.5). The goal is to relate theNoether number with Davenport constants of suitable monoids as a generalization ofthe equation b ( G ) = D ( G ) in the abelian case. Indeed, the k th Davenport constant D k ( G ) of an abelian group G is recovered as our k th Davenport constant of themonoid B ( G ) of zero-sum sequences over G .We apply the new concept of the k th Davenport constants to two classes ofBF-monoids. First, to the monoid B ( G , V ) associated to a G -module V in Subsec-tion 4.4 (when G is abelian we recover the monoid M G of G -invariant monomialsfrom Subsection 4.3), whose Davenport constants provide a lower bound for the cor-responding Noether numbers (see Proposition 4.12). Second, we study the monoidof product-one sequences over finite groups (Subsections 3.1 and 3.3). We derivea variety of features of the k th Davenport constants of the monoid of product-onesequences over G and observe that they are strikingly similar to the correspondingfeatures of the k th Noether numbers (see Subsection 5.1 for a comparison).We pose a problem on the relationship between Noether numbers and Davenportconstants of non-abelian groups (Problem 2) and we illustrate the efficiency of theabove methods by Examples 5.2, 5.3, and 5.4 (appearing for the first time), wherethe explicit value of Noether numbers and Davenport constants of some non-abeliangroups are determined. Throughout this paper, let G be a finite group, F be a field, andV be a finite dimensional F -vector space endowed with a linear action of G. We denote by N the set of positive integers, and we put N = N ∪ { } . For every n ∈ N , we denote by C n a cyclic group with n elements. For real numbers a , b ∈ R ,we set [ a , b ] = { x ∈ Z : a ≤ x ≤ b } . If A , B are sets, we write A ⊂ B to mean that A is contained in B but may be equal to B . In Subsections 2.1 – 2.4 we gather basicmaterial on Krull monoids and C-monoids. In Subsection 2.5 we introduce a newconcept, namely Davenport constants of BF-monoids. Monoids and Domains: Ideal theoretic and divisor theoreticconcepts
Our notation and terminology follows [40] and [51] (note that the monoids in[51] do contain a zero-element, whereas the monoids in [40] and in the presentmanuscript do not contain a zero-element). By a monoid , we mean a commutative, nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 5 cancellative semigroup with unit element. Then the multiplicative semigroup R • = R \ { } of non-zero elements of a domain is a monoid. Following the philosophyof multiplicative ideal theory we describe the arithmetic and the theory of divisorialideals of domains by means of their multiplicative monoids. Thus we start withmonoids.Let H be a multiplicatively written monoid. An element u ∈ H is called • invertible if there is an element v ∈ H with uv = • irreducible (or an atom ) if u is not invertible and, for all a , b ∈ H , u = ab implies a is invertible or b is invertible. • prime if u is not invertible and, for all a , b ∈ H , u | ab implies u | a or u | b .We denote by A ( H ) the set of atoms of H , by H × the group of invertible elements,and by H red = { aH × : a ∈ H } the associated reduced monoid of H . We say that H isreduced if | H × | =
1. We denote by q ( H ) a quotient group of H with H ⊂ q ( H ) , andfor a prime element p ∈ H , let v p : q ( H ) → Z be the p -adic valuation. Each monoidhomomorphism j : H → D induces a group homomorphism q ( H ) : q ( H ) → q ( D ) .For a subset H ⊂ H , we denote by [ H ] ⊂ H the submonoid generated by H , andby h H i ≤ q ( H ) the subgroup generated by H . We denote by e H = (cid:8) x ∈ q ( H ) : x n ∈ H for some n ∈ N (cid:9) the root closure of H , and by b H = (cid:8) x ∈ q ( H ) : there exists c ∈ H such that cx n ∈ H for all n ∈ N (cid:9) the complete integral closure of H . Both e H and b H are monoids, and we have H ⊂ e H ⊂ b H ⊂ q ( H ) . We say that H is root closed(completely integrally closed resp.) if H = e H ( H = b H resp.). For a set P , we denoteby F ( P ) the free abelian monoid with basis P . Then every a ∈ F ( P ) has a uniquerepresentation in the form a = (cid:213) p ∈ P p v p ( a ) , where v p ( a ) ∈ N and v p ( a ) = p ∈ P . The monoid H is said to be • atomic if every a ∈ H \ H × is a product of finitely many atoms of H . • factorial if every a ∈ H \ H × is a product of finitely many primes of H (equiva-lently, H = H × × F ( P ) where P is a set of representatives of primes of F ). • finitely generated if H = [ E ] for some finite subset E ⊂ H .If H = H × × F ( P ) is factorial and a ∈ H , then | a | = (cid:229) p ∈ P v p ( a ) ∈ N is called thelength of a . If H is reduced, then it is finitely generated if and only if it is atomicand A ( H ) is finite. Since every prime is an atom, every factorial monoid is atomic.For every non-unit a ∈ H , L H ( a ) = L ( a ) = { k ∈ N : a may be written as a product of k atoms } ⊂ N denotes the set of lengths of a . For convenience, we set L ( a ) = { } for a ∈ H × . Wesay that H is a BF-monoid if it is atomic and all sets of lengths are finite. A monoidhomomorphism j : H → D is said to be • a divisor homomorphism if j ( a ) | j ( b ) implies that a | b for all a , b ∈ H . K´alm´an Cziszter and M´aty´as Domokos and Alfred Geroldinger • cofinal if for every a ∈ D there is an a ∈ H such that a | j ( a ) . • a divisor theory (for H ) if D = F ( P ) for some set P , j is a divisor homomor-phism, and for every p ∈ P , there exists a finite nonempty subset X ⊂ H satisfying p = gcd (cid:0) j ( X ) (cid:1) .Obviously, every divisor theory is cofinal. Let H ⊂ D be a submonoid. Then H ⊂ D is called • saturated if the embedding H ֒ → D is a divisor homomorphism. • divisor closed if a ∈ H , b ∈ D and b | a implies b ∈ H . • cofinal if the embedding H ֒ → D is cofinal.It is easy to verify that H ֒ → D is a divisor homomorphism if and only if H = q ( H ) ∩ D , and if this holds, then H × = D × ∩ H . If H ⊂ D is divisor closed, then H ⊂ D is saturated.For subsets A , B ⊂ q ( H ) , we denote by ( A : B ) = { x ∈ q ( H ) : xB ⊂ A } , by A − = ( H : A ) , and by A v = ( A − ) − . A subset a ⊂ H is called an s -ideal of H if a H = a . A subset X ⊂ q ( H ) is called a fractional v -ideal (or a fractional diviso-rial ideal ) if there is a c ∈ H such that cX ⊂ H and X v = X . We denote by F v ( H ) the set of all fractional v -ideals and by I v ( H ) the set of all v -ideals of H . Fur-thermore, I ∗ v ( H ) is the monoid of v -invertible v -ideals (with v -multiplication) and F v ( H ) × = q (cid:0) I ∗ v ( H ) (cid:1) is its quotient group of fractional invertible v -ideals. Themonoid H is completely integrally closed if and only if every non-empty v -idealof H is v -invertible, and H is called v -noetherian if it satisfies the ACC (ascendingchain condition) on v -ideals. If H is v -noetherian, then H is a BF-monoid. We denoteby X ( H ) the set of all minimal nonempty prime s -ideals of H .The map ¶ : H → I ∗ v ( H ) , defined by ¶ ( a ) = aH for each a ∈ H , is a cofinaldivisor homomorphism. Thus, if H = { aH : a ∈ H } is the monoid of principalideals of H , then H ⊂ I ∗ v ( H ) is saturated and cofinal. Class groups and class semigroups
Let j : H → D be a monoid homomorphism. The group C ( j ) = q ( D ) / q ( j ( H )) is called the class group of j . For a ∈ q ( D ) , we denote by [ a ] j = a q ( j ( H )) ∈ C ( j ) the class containing a . We use additive notation for C ( j ) and so [ ] j is the zeroelement of C ( j ) .Suppose that H ⊂ D and that j = ( H ֒ → D ) . Then C ( j ) = q ( D ) / q ( H ) , and for a ∈ D we set [ a ] j = [ a ] D / H = a q ( H ) . Then D / H = { [ a ] D / H : a ∈ D } ⊂ C ( j ) is a submonoid with quotient group q ( D / H ) = C ( j ) . It is easy to check that D / H isa group if and only if H ⊂ D is cofinal. In particular, if D / H is finite or if q ( D ) / q ( H ) is a torsion group, then D / H = q ( D ) / q ( H ) . Let H be a monoid. Then H ⊂ I ∗ v ( H ) nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 7 is saturated and cofinal, and C v ( H ) = I ∗ v ( H ) / H = F v ( H ) × / q ( H ) is the v-class group of H .We will also need the concept of class semigroups which are a refinement ofordinary class groups in commutative algebra. Let D be a monoid and H ⊂ D asubmonoid. Two elements y , y ′ ∈ D are called H -equivalent, if y − H ∩ D = y ′− H ∩ D . H -equivalence is a congruence relation on D . For y ∈ D , let [ y ] DH denote thecongruence class of y , and let C ( H , D ) = { [ y ] DH : y ∈ D } and C ∗ ( H , D ) = { [ y ] DH : y ∈ ( D \ D × ) ∪ { }} . Then C ( H , D ) is a semigroup with unit element [ ] DH (called the class semigroup of H in D ) and C ∗ ( H , D ) ⊂ C ( H , D ) is a subsemigroup (called the reduced classsemigroup of H in D ). The map q : C ( H , D ) → D / H , defined by q ([ a ] DH ) = [ a ] D / H for all a ∈ D , is an epimorphism, and it is an isomorphism if and only if H ⊂ D is saturated. Krull monoids and Krull domains
Theorem 2.1.
Let H be a monoid. Then the following statements are equivalent : (a)H is v-noetherian and completely integrally closed,(b) ¶ : H → I ∗ v ( H ) is a divisor theory.(c)H has a divisor theory.(d)There is a divisor homomorphism j : H → D into a factorial monoid D.(e)H red is a saturated submonoid of a free abelian monoid. If H satisfies these conditions, then H is called a Krull monoid.Proof.
See [40, Theorem 2.4.8] or [51, Chapter 22].Let H be a Krull monoid. Then I ∗ v ( H ) is free abelian with basis X ( H ) . Let p ∈ X ( H ) . Then v p denotes the p -adic valuation of F v ( H ) × . For x ∈ q ( H ) , weset v p ( x ) = v p ( xH ) and we call v p the p -adic valuation of H . Then v : H → N ( X ( H )) , defined by v ( a ) = (cid:0) v p ( a ) (cid:1) p ∈ X ( H ) is a divisor theory and H = { x ∈ q ( H ) : v p ( x ) ≥ p ∈ X ( H ) } .If j : H → D = F ( P ) is a divisor theory, then there is an isomorphism F : I ∗ v ( H ) → D such that F ◦ ¶ = j , and it induces an isomorphism F : C v ( H ) → C ( j ) . Let D = F ( P ) be such that H red ֒ → D is a divisor theory. Then D and P are uniquelydetermined by H , K´alm´an Cziszter and M´aty´as Domokos and Alfred Geroldinger C ( H ) = C ( H red ) = D / H red is called the ( divisor ) class group of H , and its elements are called the classes of H .By definition, every class g ∈ C ( H ) is a subset of q ( D ) and P ∩ g is the set of primedivisors lying in g . We denote by C ( H ) ∗ = { [ p ] D / H red : p ∈ P } ⊂ C ( H ) the subsetof classes containing prime divisors (for more details we refer to the discussion afterDefinition 2.4.9 in [40]). Proposition 2.2.
Let H be a Krull monoid, and let j : H → D = F ( P ) be a divisorhomomorphism.1. There is a submonoid C ⊂ C ( j ) and an epimorphism C → C v ( H ) .2. Suppose that H ⊂ D is saturated and that q ( D ) / q ( H ) is a torsion group. We setD = { gcd D ( X ) : X ⊂ H finite } , and for p ∈ P define e ( p ) = min { v p ( h ) : h ∈ H with v p ( h ) > } .(a)D is a free abelian monoid with basis { p e ( p ) : p ∈ P } .(b)The embedding H ֒ → D is a divisor theory for H.Proof.
1. follows from [40, Theorem 2.4.8], and 2. from [74, Lemma 3.2].Let R be a domain with quotient field K . Then R • = R \ { } is a monoid, andall notions defined for monoids so far will be applied for domains. To mention acouple of explicit examples, we denote by q ( R ) the quotient field of R and we have q ( R ) = q ( R • ) ∪ { } , and for the complete integral closure we have b R = c R • ∪ { } (where b R is the integral closure of R in its quotient field). We denote by X ( R ) theset of all minimal nonzero prime ideals of R , by I v ( R ) the set of divisorial idealsof R , by I ∗ v ( R ) the set of v -invertible divisorial ideals of R , and by F v ( R ) theset of fractional divisorial ideals of R . Equipped with v -multiplication, F v ( R ) is asemigroup, and the map i • : F v ( R ) → F v ( R • ) , defined by a a \ { } , is a semigroup isomorphism mapping I v ( R ) onto I v ( R • ) and fractional principalideals of R onto fractional principal ideals of R • . Thus R satisfies the ACC on divi-sorial ideals of R if and only if R • satisfies the ACC on divisorial ideals of R • . Fur-thermore, R is completely integrally closed if and only if R • is completely integrallyclosed. A domain R is a Krull domain if it is completely integrally closed and satis-fies the ACC on divisorial ideals of R , and thus R is a Krull domain if and only if R • is a Krull monoid. If R is a Krull domain, we set C ( R ) = C ( R • ) . The group F v ( R ) × is the group of v -invertible fractional ideals and the set I ∗ v ( R ) = F v ( R ) × ∩ I v ( R ) of all v -invertible v -ideals of R is a monoid with quotient group F v ( R ) × . The em-bedding of the non-zero principal ideals H ( R ) ֒ → I ∗ v ( R ) is a cofinal divisor homo-morphism, and the factor group C v ( R ) = F v ( R ) × / { aR : a ∈ K × } = I ∗ v ( R ) / H ( R ) nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 9 is called the v-class group of R . The map i • induces isomorphisms F v ( R ) × ∼ → F v ( R • ) × , I ∗ v ( R ) ∼ → I ∗ v ( R • ) , and C v ( R ) ∼ → C v ( R • ) , and in the sequel we shallidentify these monoids and groups.The above correspondence between domains and their monoids of non-zero ele-ments can be extended to commutative rings with zero-divisors and their monoidsof regular elements ([43, Theorem 3.5]), and there is an analogue for prime Goldierings ([38, Proposition 5.1]). Examples 2.3.
1. (Domains) As mentioned above, the multiplicative monoid R • of a domain R isa Krull monoid if and only if R is a Krull domain. Thus Property (a) in Theorem 2.1implies that a noetherian domain is Krull if and only if it is normal (i.e. integrallyclosed in its field of fractions). In particular, rings of invariants are Krull, as we shallsee in Theorem 4.1.2. (Submonoids of domains) Regular congruence submonoids of Krull domainsare Krull ([40, Proposition 2.11.6].3. (Monoids of modules) Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right) R -modules which is closed under finite direct-sums, direct summands, and isomorphisms. Then the set V ( C ) of isomorphismclasses of modules is a commutative semigroup with operation induced by the directsum. If the endomorphism ring of each module in C is semilocal, then V ( C ) is aKrull monoid ([19, Theorem 3.4]). For more information we refer to [20, 21, 1].4. (Monoids of product-one sequences) In Theorem 3.2 we will characterize themonoids of product-one sequences which are Krull. C-monoids and C-domains
A monoid H is called a C- monoid if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and the reduced class semigroup C ∗ ( H , F ) is finite. Adomain is called a C- domain if R • is a C-monoid. Proposition 2.4.
Let F be a factorial monoid and H ⊂ F a submonoid such thatH ∩ F × = H × .1. If H is a C -monoid, then H is v-noetherian with ( H : b H ) = /0 , and the completeintegral closure b H is a Krull monoid with finite class group C ( b H ) .2. Suppose that F / F × is finitely generated, say F = F × × [ p , . . . , p s ] with pair-wise non-associated prime elements p , . . . , p s . Then the following statementsare equivalent : (a) H is a C -monoid defined in F. (b) There exist some a ∈ N and a subgroup W ≤ F × such that ( F × : W ) | a , W ( H \ H × ) ⊂ H, and for all j ∈ [ , s ] and a ∈ p a j F we have a ∈ H if and only ifp a j a ∈ H.Proof.
For 1., see [40, Theorems 2.9.11 and 2.9.13] and for 2. see [40, Theorems2.9.7].
Examples 2.5.
1. (Krull monoids) A Krull monoid is a C-monoid if and only if the class groupis finite ([40, Theorem 2.9.12]).2. (Domains) Let R be a domain. Necessary conditions for R being a C-domainare given in Proposition 2.4. Thus suppose that R is a Mori domain (i.e., a v -noetherian domain) with nonzero conductor f = ( R : b R ) and suppose that C ( b R ) isfinite. If R / f is finite, then R is a C-domain by [40, Theorem 2.11.9]. This resultgeneralizes to rings with zero-divisors ([43]), and in special cases we know that R is a C-domain if and only if R / f is finite ([69]).3. (Congruence monoids) Let R be Krull domain with finite class group C ( R ) and H ⊂ R a congruence monoid such that R / f is finite where f is an ideal of definitionfor H . If R is noetherian or f is divisorial, then H is a C-monoid ([40, Theorem2.11.8]). For a survey on arithmetical congruence monoids see [2].4. In Subsection 3.1 we shall prove that monoids of product-one sequences areC-monoids (Theorem 3.2), and we will meet C-monoids again in Proposition 4.11dealing with the monoid B ( G , V ) .Finitely generated monoids allow simple characterizations when they are Krull orwhen they are C-monoids. We summarize these characterizations in the next lemma. Proposition 2.6.
Let H be a monoid such that H red is finitely generated.1. Then H is v-noetherian with ( H : b H ) = /0 , e H = b H, e H / H × is finitely generated,and b H is a Krull monoid. In particular, H is a Krull monoid if and only if H = b H.2. H is a C -monoid if and only if C ( b H ) is finite.3. Suppose that H is a submonoid of a factorial monoid F = F × × F ( P ) . Then thefollowing statements are equivalent : a. H is a C -monoid defined in F, F × / H × is a torsion group, and for every p ∈ Pthere is an a ∈ H such that v p ( a ) > .b. For every a ∈ F, there is an n a ∈ N with a n a ∈ H.If ( a ) and ( b ) hold, then P is finite and e H = b H = q ( H ) ∩ F ⊂ F is saturated andcofinal.Proof.
1. follows from [40, 2.7.9 - 2.7.13], and 2. follows from [41, Proposition4.8].3. (a) ⇒ (b) For every p ∈ P , we set d p = gcd (cid:0) v p ( H ) (cid:1) , and by assumption wehave d p >
0. We set P = { p d p : p ∈ P } and F = F × × F ( P ) . By [40, Theorem nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 11 H is a C-monoid defined in F and there is a divisor theory ¶ : b H → F ( P ) .By construction of F , it is sufficient to prove the assertion for all a ∈ F . Since F × / H × is a torsion group, it is sufficient to prove the assertion for all a ∈ F ( P ) .Let a ∈ F ( P ) . Since C ( b H ) is finite, there is an n ′ a ∈ N such that a n ′ a ∈ b H . Since b H = e H , there is an n ′′ a ∈ N such that ( a n ′ a ) n ′′ a ∈ H .(b) ⇒ (a) For every p ∈ P there is an n p ∈ N such that p n p ∈ H whence v p ( p n p ) = n p >
0. Clearly, we have b H ⊂ b F = F and hence b H ⊂ q ( b H ) ∩ F = q ( H ) ∩ F . Since foreach a ∈ F there is an n a ∈ N with a n a ∈ H , we infer that q ( H ) ∩ F ⊂ e H = b H andhence b H = q ( H ) ∩ F . Furthermore, H ⊂ F and b H ⊂ F are cofinal, and q ( F ) / q ( H ) = F / H is a torsion group. Clearly, q ( H ) ∩ F ⊂ F is saturated, and thus b H is Krull.Since b H × = b H ∩ F × and H × = b H × ∩ H , it follows that H × = H ∩ F × and then weobtain that F × / H × is a torsion group.By 1., b H / H × is finitely generated, say b H / H × = { u H × , . . . , u n H × } , and set P = { p ∈ P : p divides u · . . . · u n in F } . Then P is finite, and we assert that P = P .If there would exist some p ∈ P \ P , then there is an n p ∈ N such that p n p ∈ H and hence p n p H × is a product of u H × , . . . , u n H × , a contradiction. Therefore P isfinite, F / F × is a finitely generated monoid, q ( F ) / F × is a finitely generated group,and therefore q ( F ) / q ( H ) F × is a finitely generated torsion group and thus finite.Since j : b H → F → F / F × is a divisor homomorphism and C ( j ) = q ( F ) / q ( H ) F × ,Proposition 2.2.1 implies that C ( b H ) is an epimorphic image of a submonoid of q ( F ) / q ( H ) F × and thus C ( b H ) is finite. Thus 2. implies that H is a C-monoid (in-deed, Property 2.(b) of Proposition 2.4 holds and hence H is a C-monoid defined in F ). Davenport constants of BF-monoids
Let H be a BF-monoid. For every k ∈ N , we study the sets M k ( H ) = { a ∈ H : max L ( a ) ≤ k } and M k ( H ) = { a ∈ H : max L ( a ) = k } . A monoid homomorphism | · | : H → ( N , +) will be called a degree function on H . In this section we study abstract monoids having a degree function. The resultswill be applied in particular to monoids of product-one sequences and to monoids B ( G , V ) (see Subsections 3.3 and 4.4). In all our applications the monoid H will bea submonoid of a factorial monoid F and if not stated otherwise the degree functionon H will be the restriction of the length function on F .If q : H → B is a homomorphism and H and B have degree functions, then wesay that q is degree preserving if | a | H = | q ( a ) | B for all a ∈ H . Suppose we are givena degree function on H and k ∈ N , then D k ( H ) = sup {| a | : a ∈ M k ( H ) } ∈ N ∪ { ¥ } is called the large kth Davenport constant of H (with respect to | · | H ). Clearly, M ( H ) = A ( H ) ∪ H × . We call D ( H ) = D ( H ) = sup {| a | : a ∈ A ( H ) } ∈ N ∪ { ¥ } the Davenport constant of H . For every k ∈ N , we have M k ( H ) ⊂ M k + ( H ) , D k ( H ) ≤ D k + ( H ) , and D k ( H ) ≤ k D ( H ) . Furthermore, we have | u | = u ∈ H × . Therefore the degree function on H induces automatically a degreefunction | · | : H red → ( N , +) , and so the k th Davenport constant of H red is defined.Obviously we have D k ( H ) = D k ( H red ) . Let e ( H ) denote the smallest ℓ ∈ N ∪ { ¥ } with the following property:There is a K ∈ N such that every a ∈ H with | a | ≥ K is divisible by an element b ∈ H \ H × with | b | ≤ ℓ .Clearly, e ( H ) ≤ D ( H ) . Proposition 2.7.
Let H be a BF -monoid and | · | : H → ( N , +) be a degree function.1. If H red is finitely generated, then the sets M k ( H red ) are finite and D k ( H ) < ¥ forevery k ∈ N .2. If D ( H ) < ¥ , then there exist constants D H , K H ∈ N such that D k ( H ) = k e ( H ) + D H for all k ≥ K H .3. If D ( H ) < ¥ , then the map N → Q , k D k ( H ) k is non-increasing.4. Suppose that H has a prime element. Then D k ( H ) = max (cid:8) | a | : a ∈ M k ( H ) (cid:9) ≤ k D ( H ) andk D ( H ) = max (cid:8) | a | : a ∈ H , min L ( a ) ≤ k (cid:9) = max (cid:8) | a | : a ∈ H , k ∈ L ( a ) (cid:9) . Proof.
1. Suppose that H red is finitely generated. Then A ( H red ) is finite whence M k ( H ) is finite for every k ∈ N . It follows that D ( H ) < ¥ and D k ( H ) ≤ k D ( H ) < ¥ for all k ∈ N .2. Suppose that D ( H ) < ¥ and note that e ( H ) ≤ D ( H ) . Let f ( H ) ∈ N be thesmallest K ∈ N such that every a ∈ H with | a | ≥ K is divisible by an element b ∈ H with | b | ≤ e ( H ) . We define A = { a ∈ A ( H ) : | a | = e ( H ) } . Let k ∈ N and continuewith the following assertion. A. There exist a , . . . , a k ∈ A such that a . . . a k ∈ M k ( H ) . In particular, D k ( H ) ≥| a . . . a k | = k e ( H ) . Proof of A . Assume to the contrary that for all a , . . . , a k ∈ A we have max L ( a ) > k . Thus the product a . . . a k is divisible by an atom u ∈ A ( H ) with | u | < e ( H ) . Weset K = f ( H ) + ( k − ) e ( H ) and choose a ∈ H with | a | ≥ K . Then a can be writtenin the form a = a . . . a k b where a , . . . , a k , b ∈ H and | a i | ≤ e ( H ) for all i ∈ [ , k ] . Ifthere is some i ∈ [ , k ] with | a i | < e ( H ) , then a i is a divisor of a with | a i | < e ( H ) .Otherwise, a , . . . , a k ∈ A and by our assumption the product a . . . a k and hence a has a divisor of degree strictly smaller than e ( H ) . This is a contradiction to thedefinition of e ( H ) . ⊓⊔ (Proof of A ) nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 13 Now let k ≥ f ( H ) / e ( H ) −
1. Then A implies that D k ( H ) + e ( H ) ≥ ( k + ) e ( H ) ≥ f ( H ) . Let a ∈ H with | a | > D k ( H ) + e ( H ) . Then, by definition of f ( H ) , there are b , c ∈ H such that a = bc with | c | ≤ e ( H ) and hence | b | > D k ( H ) . This implies thatmax L ( b ) > k , whence max L ( a ) > k + a / ∈ M k + ( H ) . Therefore we obtainthat D k + ( H ) ≤ D k ( H ) + e ( H ) and thus0 ≤ D k + ( H ) − ( k + ) e ( H ) ≤ D k ( H ) − k e ( H ) . Since a non-increasing sequence of non-negative integers stabilizes, the assertionfollows.3. Suppose that D ( H ) < ¥ . Let k ∈ N , a ∈ M k + ( H ) with | a | = D k + ( H ) ,and set l = max L ( a ) . Then l ≤ k +
1. If l ≤ k , then a ∈ M k ( H ) and D k + ( H ) ≥ D k ( H ) ≥ | a | = D k + ( H ) whence D k ( H ) = D k + ( H ) . Suppose that l = k +
1. We set a = a . . . a k + with a , . . . , a k + ∈ A ( H ) and | a | ≥ . . . ≥ | a k + | whence | a k + | ≤ ( | a | + . . . + | a k | ) / k . It follows that D k + ( H ) k + = | a | + · · · + | a k + | k + ≤ | a | + · · · + | a k | k ≤ D k ( H ) k , where the last inequality holds because a . . . a k ∈ M k ( H ) .4. Let p ∈ H be a prime element. We assert that D k ( H ) ≤ max (cid:8) | a | : a ∈ H , max L ( a ) = k (cid:9) . ( ∗ )Indeed, if a ∈ M k ( H ) and max L ( a ) = l ≤ k , then ap k − l ∈ M k ( H ) and | a | ≤ | ap k − l | ≤ max (cid:8) | a | : a ∈ H , max L ( a ) = k (cid:9) , and hence ( ∗ ) follows. Next we assert thatmax (cid:8) | a | : a ∈ H , min L ( a ) ≤ k (cid:9) ≤ k D ( H ) . ( ∗∗ )Let a ∈ H with min L ( a ) = l ≤ k , say a = u . . . u l , where u , . . . , u l ∈ A ( H ) . Then | a | = | u | + . . . + | u l | ≤ l D ( H ) ≤ k D ( H ) , and thus ( ∗∗ ) follows. Using ( ∗ ) and ( ∗∗ ) we infer that D k ( H ) ≤ max (cid:8) | a | : a ∈ H , max L ( a ) = k (cid:9) ≤ max (cid:8) | a | : a ∈ H , max L ( a ) ≤ k (cid:9) = D k ( H ) ≤ max (cid:8) | a | : a ∈ H , min L ( a ) ≤ k (cid:9) and that k D ( H ) = max (cid:8) | a | : a ∈ H , k ∈ L ( a ) (cid:9) ≤ max (cid:8) | a | : a ∈ H , min L ( a ) ≤ k (cid:9) ≤ k D ( H ) . Let F be a factorial monoid and H ⊂ F a submonoid such that H × = H ∩ F × .Then H is a BF-monoid by [40, Corollary 1.3.3]. For k ∈ N , let M ∗ k ( H ) denote theset of all a ∈ F such that a is not divisible by a product of k non-units of H . Therestriction of the usual length function | · | : F → N on F (introduced in Subsec- tion 2.1) gives a degree function on H . We define the small kth Davenport constant d k ( H ) as d k ( H ) = sup {| a | : a ∈ M ∗ k ( H ) } ∈ N ∪ { ¥ } . (1)In other words, 1 + d k ( H ) is the smallest integer ℓ ∈ N such that every a ∈ F oflength | a | ≥ ℓ is divisible by a product of k non-units of H . We call d ( H ) = d ( H ) the small Davenport constant of H . Clearly we have M ∗ k ( H ) ⊂ M ∗ k + ( H ) hence d k ( H ) ≤ d k + ( H ) .Furthermore, let h ( H ) denote the smallest integer ℓ ∈ N ∪ { ¥ } such that every a ∈ F with | a | ≥ ℓ has a divisor b ∈ H \ H × with | b | ∈ [ , e ( H )] . For p ∈ A ( F ) denote by o p the smallest integer ℓ ∈ N ∪ { ¥ } such that p o p ∈ H . Clearly, we have o p ≤ h ( H ) for all p ∈ A ( F ) . Proposition 2.8.
Let F = F × × F ( P ) be a factorial monoid and H ⊂ F a sub-monoid such that H × = H ∩ F × , and let k ∈ N .1. If for every a ∈ F there is a prime p ∈ F such that ap ∈ H, then + d k ( H ) ≤ D k ( H ) .2. Suppose that H red is finitely generated and that for every a ∈ F there is an n a ∈ Hsuch that a n a ∈ H. Then H is a C -monoid and we have(a) e ( H ) = max { o p : p ∈ P } and h ( H ) < ¥ .(b) d k ( H ) + ≥ k e ( H ) and there exist constants d H ∈ Z ≥− , k H ∈ N such that d k ( H ) = k e ( H ) + d H for all k ≥ k H .Proof.
1. Let a ∈ M ∗ k ( H ) such that | a | = d k ( H ) . We choose a prime p ∈ F such that ap ∈ H . Take any factorization ap = u . . . u ℓ where u i ∈ A ( H ) . We may assumethat p | u in F . Then u . . . u ℓ | a in F and hence ℓ − < k . Thus it follows that ap ∈ M k ( H ) and D k ( H ) ≥ | ap | = | a | + ≥ d k ( H ) + H is a C-monoid, P is finite and hence e ( H ) < ¥ . If p ∈ P , then p o p ∈ A ( H ) and by the minimality of o p , p o p does not have a divisor b ∈ H \ H × such that | b | < o p . Thus it follows that e ( H ) ≥ max { o p : p ∈ P } . For thereverse inequality, note that by Proposition 2.4.2 there exists an a ∈ N such that forall p ∈ P and all a ∈ p a F we have a ∈ H if and only if p a a ∈ H . Since any multipleof a has the same property, we may assume that a is divisible by o p for all p ∈ P .Let b ∈ H with | b | > | P | ( a − ) . Then there exists a p ∈ P such that b ∈ p a F ∩ H .Hence b is divisible in H by p a , implying in turn that p o p ∈ A ( H ) divides b in H .Therefore we obtain that e ( H ) ≤ max { o p : p ∈ P } .If a ∈ F with | a | ≥ (cid:229) p ∈ P ( o p − ) , then there is a p ∈ P such that p o p divides a in F , and thus h ( H ) ≤ + (cid:229) p ∈ P ( o p − ) .2.(b) Let p ∈ P with o ( p ) = e ( H ) . Then p ko p − ∈ M ∗ k ( H ) and | p ko p − | = k e ( H ) −
1, showing the inequality d k ( H ) + ≥ k e ( H ) for all k ∈ N . Now let k ∈ N be suchthat 1 + d k ( H ) + e ( H ) ≥ h ( H ) , and let a ∈ F with | a | ≥ d k ( H ) + e ( H ) +
1. Then, bydefinition of h ( H ) , there are b ∈ F and c ∈ H \ H × such that a = bc with | c | ≤ e ( H ) and | b | > d k ( H ) . This implies that b is divisible by a product of k non-units of H whence a is divisible by a product of k + H . Therefore it follows that1 + d k + ( H ) ≤ d k ( H ) + e ( H ) + nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 15 ≤ d k + ( H ) − k e ( H ) ≤ d k ( H ) − ( k − ) e ( H ) for all sufficiently large k . Since a non-increasing sequence of non-negative integers stabilizes, the assertionfollows.
This section is devoted to Zero-Sum Theory, a vivid subfield of Arithmetic Combina-torics (see [32, 37, 49]). In Subsection 3.1 we give an algebraic study of the monoidof product-one sequences over finite but not necessarily abelian groups. In Subsec-tion 3.2 we put together well-known material on transfer homomorphisms used inSubsections 4.2 and 4.3. In Subsections 3.3 and 3.4 we consider the k th Davenportconstants of finite groups. In particular, we gather results which will be needed inSubsection 5.2 and results having relevance in invariant theory by Proposition 4.7. The monoid of product-one sequences
Let G ⊂ G be a subset and let G ′ = [ G , G ] = h g − h − gh : g , h ∈ G i denote thecommutator subgroup of G . A sequence over G means a finite sequence of termsfrom G which is unordered and repetition of terms is allowed, and it will be con-sidered as an element of the free abelian monoid F ( G ) . In order to distinguishbetween the group operation in G and the operation in F ( G ) , we use the sym-bol · for the multiplication in F ( G ) , hence F ( G ) = (cid:0) F ( G ) , · (cid:1) —this coincideswith the convention in the monographs [40, 49]–and we denote multiplication in G by juxtaposition of elements. To clarify this, if S , S ∈ F ( G ) and g , g ∈ G ,then S · S ∈ F ( G ) has length | S | + | S | , S · g ∈ F ( G ) has length | S | + g · g ∈ F ( G ) is a sequence of length 2, but g g is an element of G . Further-more, in order to avoid confusion between exponentiation in G and exponentia-tion in F ( G ) , we use brackets for the exponentiation in F ( G ) . So for g ∈ G , S ∈ F ( G ) , and k ∈ N , we have g [ k ] = g · . . . · g | {z } k ∈ F ( G ) with | g [ k ] | = k , and S [ k ] = S · . . . · S | {z } k ∈ F ( G ) . Now let S = g · . . . · g ℓ = (cid:213) g ∈ G g v g ( S ) , be a sequence over G (in this notation, we tacitly assume that ℓ ∈ N and g , . . . , g ℓ ∈ G ). Then | S | = ℓ = S = F ( G ) is the identity element in F ( G ) , and then S will also be called the trivial sequence . The elements in F ( G ) \ { F ( G ) } are called nontrivial sequences . We use all notions of divisibility theory in generalfree abelian monoids. Thus, for an element g ∈ G , we refer to v g ( S ) as the multi-plicity of g in S . A divisor T of S will also be called a subsequence of S . We callsupp ( S ) = { g , . . . , g ℓ } ⊂ G the support of S . When G is written multiplicatively(with unit element 1 G ∈ G ), we use p ( S ) = { g t ( ) . . . g t ( ℓ ) ∈ G : t a permutation of [ , ℓ ] } ⊂ G to denote the set of products of S (if | S | =
0, we use the convention that p ( S ) = { G } ). Clearly, p ( S ) is contained in a G ′ -coset. When G is written additively withcommutative operation, we likewise let s ( S ) = g + . . . + g ℓ ∈ G denote the sum of S . Furthermore, we denote by S ( S ) = { s ( T ) : T | S and 1 = T } ⊂ G and P ( S ) = [ T | S = T p ( T ) ⊂ G , the subsequence sums and subsequence products of S . The sequence S is called • a product-one sequence if 1 G ∈ p ( S ) , • product-one free if 1 G / ∈ P ( S ) .Every map of finite groups j : G → G extends to a homomorphism j : F ( G ) → F ( G ) where j ( S ) = j ( g ) · . . . · j ( g ℓ ) . If j is a group homomorphism, then j ( S ) is a product-one sequence if and only if p ( S ) ∩ Ker ( j ) = /0. We denote by B ( G ) = { S ∈ F ( G ) : 1 G ∈ p ( S ) } the set of all product-one sequences over G , and clearly B ( G ) ⊂ F ( G ) is asubmonoid. We will use all concepts introduced in Subsection 2.5 for the monoid B ( G ) with the degree function stemming from the length function on the freeabelian monoid F ( G ) . For all notations ∗ ( H ) introduced for a monoid H we write– as usual – ∗ ( G ) instead of ∗ ( B ( G )) . In particular, for k ∈ N , we set M k ( G ) = M k ( B ( G )) , D k ( G ) = D k ( B ( G )) , h ( G ) = h ( B ( G )) , e ( G ) = e ( B ( G )) , andso on. By Proposition 2.8.2(a), e ( G ) = max { ord ( g ) : g ∈ G } . Note that M ∗ ( G ) is the set of all product-one free sequences over G . In particular, D ( G ) = sup {| S | : S ∈ A ( G ) } ∈ N ∪ { ¥ } is the large Davenport constant of G , and d ( G ) = sup {| S | : S ∈ F ( G ) is product-one free } ∈ N ∪ { ¥ } is the small Davenport constant of G . Their study will be the focus of the Subsec-tions 3.3 and 3.4. nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 17 Lemma 3.1.
Let G ⊂ G be a subset.1. B ( G ) ⊂ F ( G ) is a reduced finitely generated submonoid, A ( G ) is finite, and D ( G ) ≤ | G | . Furthermore, M k ( G ) is finite and D k ( G ) < ¥ for all k ∈ N .2. Let S ∈ F ( G ) be product-one free.a. If g ∈ p ( S ) , then g − · S ∈ A ( G ) . In particular, d ( G ) + ≤ D ( G ) .b. If | S | = d ( G ) , then P ( S ) = G \ { G } and hence d ( G ) = max {| S | : S ∈ F ( G ) with P ( S ) = G \ { G }} .3. If G is cyclic, then d ( G ) + = D ( G ) = | G | .Proof.
1. We assert that for every U ∈ A ( G ) we have | U | ≤ | G | . Then A ( G ) ⊂ A ( G ) is finite and D ( G ) ≤ D ( G ) ≤ | G | . As already mentioned, B ( G ) ⊂ F ( G ) is a submonoid, and clearly B ( G ) × = { F ( G ) } . Since F ( G ) is factorial and B ( G ) × = B ( G ) ∩ F ( G ) × , B ( G ) is atomic by [40, Corollary 1.3.3]. Thismeans that B ( G ) = [ A ( G ) ∪ B ( G ) × ] , and thus B ( G ) is finitely generated.Since B ( G ) is reduced and finitely generated, the sets M k ( G ) are finite by Propo-sition 2.7.Now let U ∈ B ( G ) , say U = g · . . . · g ℓ with g g . . . g ℓ = G . We suppose that ℓ > | G | and show that U / ∈ A ( G ) . Consider the set M = { g g . . . g i : i ∈ [ , ℓ ] } . Since ℓ > | G | , there are i , j ∈ [ , ℓ ] with i < j and g . . . g i = g . . . g j . Then g i + . . . g j = G and thus g . . . g i g j + . . . g ℓ = G which implies that U is the product of two non-trivial product-one subsequences.2.(a) If g ∈ p ( S ) , then S can be written as S = g · . . . · g ℓ such that g = g . . . g ℓ ,which implies that g − · g · . . . · g ℓ ∈ A ( G ) .2.(b) If S is product-one free with | S | = d ( G ) , and if there would be an h ∈ G \ { P ( S ) ∪ { G }} , then T = h − · S would be product-one free of length | T | = | S | + > d ( G ) , a contradiction. Thus every product-one free sequence S of length | S | = d ( G ) satisfies P ( S ) = G \ { G } . If S is a sequence with P ( S ) = G \ { G } , then S is product-one free and hence | S | ≤ d ( G ) .3. Clearly, the assertion holds for | G | =
1. Suppose that G is cyclic of order n ≥ g ∈ G with ord ( g ) = n . Then g [ n − ] is product-one free, and thus 1. and 2.imply that n ≤ + d ( G ) ≤ D ( G ) ≤ n .The next result gathers the algebraic properties of monoids of product-one se-quences and highlights the difference between the abelian and the non-abelian case. Theorem 3.2.
Let G ⊂ G be a subset and let G ′ denote the commutator subgroupof h G i .1. B ( G ) ⊂ F ( G ) is cofinal and B ( G ) is a finitely generated C -monoid. ^ B ( G ) = \ B ( G ) is a finitely generated Krull monoid, the embedding \ B ( G ) ֒ → F ( G ) isa cofinal divisor homomorphism with class group F ( G ) / B ( G ) , and the map F : F ( G ) / B ( G ) −→ h G i / G ′ [ S ] F ( G ) / B ( G ) gG ′ for any g ∈ p ( S ) is a group epimorphism. Suppose that G = G. Then F is an isomorphism, everyclass of C ( \ B ( G )) contains a prime divisor, and if | G | 6 = , then \ B ( G ) ֒ → F ( G ) is a divisor theory.2. The following statements are equivalent : (a) B ( G ) is a Krull monoid.(b) B ( G ) is root closed.(c) B ( G ) ⊂ F ( G ) is saturated.3. B ( G ) is a Krull monoid if and only if G is abelian.4. B ( G ) is factorial if and only if | G | ≤ .Proof. B ( G ) is finitely generated by Lemma 3.1. If n = lcm { ord ( g ) : g ∈ G } ,then S [ n ] ∈ B ( G ) for each S ∈ F ( G ) . Thus B ( G ) ⊂ F ( G ) and \ B ( G ) ֒ → F ( G ) are cofinal, F ( G ) / B ( G ) is a group and F ( G ) / B ( G ) = q (cid:0) F ( G ) (cid:1) / q (cid:0) B ( G ) (cid:1) = q (cid:0) F ( G ) (cid:1) / q (cid:0) \ B ( G ) (cid:1) is the class group of the embedding \ B ( G ) ֒ → F ( G ) . All statements on the struc-ture of B ( G ) and \ B ( G ) follow from Proposition 2.6.3, and it remains to show theassertions on F .Let S , S ′ ∈ F ( G ) , g ∈ p ( S ) , g ′ ∈ p ( S ′ ) , and B ∈ B ( G ) . Then p ( S ) ⊂ gG ′ , p ( S ′ ) ⊂ g ′ G ′ , p ( B ) ⊂ G ′ , and p ( S · B ) ⊂ gG ′ . We use the abbreviation [ S ] =[ S ] F ( G ) / B ( G ) , and note that [ S ] = [ S ′ ] if and only if there are C , C ′ ∈ B ( G ) suchthat S · C = S ′ · C ′ .In order to show that F is well-defined, suppose that [ S ] = [ S ′ ] and that S · C = S · C ′ with C , C ′ ∈ B ( G ) . Then p ( S · C ) = p ( S ′ · C ′ ) ⊂ gG ′ ∩ g ′ G ′ and hence gG ′ = g ′ G ′ . In order to show that F is surjective, let g ∈ h G i be given. Clearly, there is an S ∈ F ( G ) such that g ∈ p ( S ) whence F ([ S ]) = gG ′ .Suppose that G = G . First we show that F is injective. Let S , S ′ ∈ F ( G ) with g ∈ p ( S ) , g ′ ∈ p ( S ′ ) such that gG ′ = g ′ G ′ . Then there are k ∈ N , a , b , . . . , a k , b k ∈ G such that gg ′− = k (cid:213) i = ( a − i b − i a i b i ) . We define T = (cid:213) ki = ( a − i · b − i · a i · b i ) and obtain that S · ( S ′ · g − · T ) = S ′ · ( S · g − · T ) ∈ F ( G ) . Since 1 ∈ p ( T ) and gg ′− ∈ p ( T ) , it follows that 1 ∈ p ( S ′ · g − · T ) and 1 ∈ p ( S · g − · T ) which implies that [ S ] = [ S ′ ] . nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 19 If | G | ≤
2, then 4. will show that B ( G ) is factorial and clearly the trivial classcontains a prime divisor. Suppose that | G | ≥
3. In order to show that \ B ( G ) ֒ → F ( G ) is a divisor theory, let g ∈ G \ { G } be given. Then there is an h ∈ G \ { g − , G } , U = g · g − ∈ A ( G ) ⊂ \ B ( G ) , U ′ = g · h · ( h − g − ) ∈ A ( G ) ⊂ \ B ( G ) , and g = gcd F ( G ) ( U , U ′ ) . Thus \ B ( G ) ֒ → F ( G ) is a divisor theory.Let S ∈ F ( G ) with g ∈ p ( S ) . Then g ∈ F ( G ) is a prime divisor and we showthat [ g ] = [ S ] . Indeed, if g = G , then S ∈ B ( G ) , 1 G ∈ B ( G ) , S · G = g · S whence [ g ] = [ S ] . If ord ( g ) = n ≥
2, then g [ n ] ∈ B ( G ) , S · g [ n − ] ∈ B ( G ) , S · g [ n ] = g · S · g [ n − ] whence [ S ] = [ g ] .2. (a) ⇒ (b) Every Krull monoid is completely integrally closed and hence rootclosed.(b) ⇒ (c) Let S , T ∈ B ( G ) with T | S in F ( G ) , say S = T · U where U = g · . . . · g ℓ ∈ F ( G ) . If n = lcm (cid:0) ord ( g ) , . . . , ord ( g ℓ ) (cid:1) , then ( T [ − ] · S ) [ n ] = U [ n ] ∈ B ( G ) . Since B ( G ) is root closed, this implies that U = T [ − ] · S ∈ B ( G ) andhence T | S in B ( G ) .(c) ⇒ (a) Since F ( G ) is free abelian, B ( G ) is Krull by Theorem 2.1.3. If G is a abelian, then it is obvious that B ( G ) ⊂ F ( G ) is saturated, and thus B ( G ) is a Krull monoid by 2. Suppose that G is not abelian. Then there are g , h ∈ G with gh = hg . Then ghg − = h , S = g · h · g − · ( ghg − ) − ∈ B ( G ) , T = g · g − ∈ B ( G ) divides S in F ( G ) but T [ − ] · S = h · ( ghg − ) − does not have product-one.Thus B ( G ) ⊂ F ( G ) is not saturated and hence B ( G ) is not Krull by 2.4. If G = { } , then B ( G ) = F ( G ) is factorial. If G = { , g } , then A ( G ) = { , g [ ] } , each atom is a prime, and B ( G ) is factorial. Conversely, suppose that B ( G ) is factorial. Then B ( G ) is a Krull monoid by [40, Corollary 2.3.13], andhence G is abelian by 3. Suppose that | G | ≥
3. We show that B ( G ) is not factorial.If there is an element g ∈ G with ord ( g ) = n ≥
3, then U = g [ n ] , − U = ( − g ) [ n ] , W =( − g ) · g ∈ A ( G ) , and U · ( − U ) = W [ n ] . Suppose there is no g ∈ G with ord ( g ) ≥ e , e ∈ G with ord ( e ) = ord ( e ) = e + e =
0. Then U = e · e · ( e + e ) , W = e [ ] , W = e [ ] , W = ( e + e ) [ ] ∈ A ( G ) , and U [ ] = W · W · W .For a subset G ⊂ G , the monoid B ( G ) may be Krull or just seminormal but itneed not be Krull. We provide examples for both situations. Proposition 3.3.
Let G ⊂ G be a subset satisfying the following property P : P. For each two elements g , h ∈ G , h h i ⊂ h g , h i is normal or h g i ⊂ h g , h i is normal.Then B ( G ) is a Krull monoid if and only if h G i is abelian.Proof. If h G i is a abelian, then it is obvious that B ( G ) ⊂ F ( G ) is saturated, andthus B ( G ) is Krull by Theorem 3.2.2.Conversely, suppose that B ( G ) is Krull and that G satisfies Property P . Inorder to show that h G i is abelian, it is sufficient to prove that gh = hg for each twoelements g , h ∈ G . Let g , h ∈ G be given such that h h i ⊂ h g , h i is normal, ord ( g ) = m , ord ( h ) = n , and assume to the contrary that ghg − = h . Since g h h i g − = h h i , it follows that ghg − = h n for some n ∈ [ , n − ] . Thus ghg m − h n − n = S = g · h · g [ m − ] · h [ n − n ] ∈ B ( G ) . Clearly, T = g [ m ] ∈ B ( G ) but S · T [ − ] = h [ n − n + ] / ∈ B ( G ) .Thus B ( G ) ⊂ F ( G ) is not saturated, a contradiction. Proposition 3.4.
Let G = D n be the dihedral group, say G = h a , b i = { , a , . . . , a n − , b , ab , . . . , a n − b } , where ord ( a ) = n ≥ , ord ( b ) = , and set G = { ab , b } . Then B ( G ) is a Krull monoid if and only if n is even.Proof. Clearly, we have ord ( ab ) = ord ( b ) = h G i = G . Suppose that n isodd and consider the sequence S = ( ab ) [ n ] · b [ n ] . Since (cid:0) ( ab ) b (cid:1) n =
1, it follows that S is a product-one sequence. Obviously, S = ( ab ) [ n − ] · b [ n − ] ∈ B ( G ) and S =( ab ) · b / ∈ B ( G ) . Since S = S · S , it follows that B ( G ) ⊂ F ( G ) is not saturated,and hence B ( G ) is not Krull by Theorem 3.2.2.Suppose that n is even. Then A ( G ) = { ( ab ) [ ] , b [ ] } and B ( G ) = { ( ab ) [ ℓ ] · b [ m ] : ℓ, m ∈ N even } . This description of B ( G ) implies immediately that B ( G ) ⊂ F ( G ) is saturated, and hence B ( G ) is Krull by Theorem 3.2.2. Remark. ( Seminormality of B ( G ) ) A monoid H is called seminormal if for all x ∈ q ( H ) with x , x ∈ H it follows that x ∈ H . Thus, by definition, every root closedmonoid is seminormal.1. Let n ≡ G = D n the dihedral group, say G = h a , b i = { , a , . . . , a n − , b , ab , . . . , a n − b } , where ord ( a ) = n , ord ( b ) =
2, and a k ba l b = a k − l for all k , l ∈ Z . We consider the sequence S = a (cid:2) n − (cid:3) · b [ ] ∈ F ( G ) . Then S [ ] = (cid:0) a (cid:2) n − (cid:3) · b · a (cid:2) n − (cid:3) · b (cid:1) · ( b · b ) and S [ ] = a [ n ] · (cid:0) a (cid:2) n − (cid:3) · b · a (cid:2) n − (cid:3) · b (cid:1) · b [ ] are both in B ( G ) whence S ∈ q (cid:0) B ( G ) (cid:1) , but obviously S / ∈ B ( G ) . Thus B ( { a , b } ) and B ( G ) are not seminormal.2. Let G = H = { E , I , J , K , − E , − I , − J , − K } be the quaternion group with therelations IJ = − JI = K , JK = − KJ = I , and KI = − IK = J , and set G = { I , J } . By Theorem 3.2, B ( G ) is not Krull and by Proposition 3.3, B ( G ) is not Krull. However, we assert that B ( G ) is seminormal.First, we are going to derive an explicit description of B ( G ) . Since E =( − E )( − E ) = ( KK )( II ) = ( IJ )( IJ )( II ) , it follows that U = I [ ] · J [ ] ∈ B ( G ) . As-sume that U = U · U with U , U ∈ A ( G ) and | U | ≤ | U | . Then | U | ∈ { , } , but U does not have a subsequence with product one and length two or three. Thus U ∈ nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 21 A ( G ) and similarly we obtain that I [ ] · J [ ] ∈ A ( G ) . Since D ( G ) ≤ D ( G ) =
6, itis easy to check that A ( G ) = { I [ ] , J [ ] , I [ ] · J [ ] , I [ ] · J [ ] , I [ ] · J [ ] } . This implies that B ( G ) = { I [ k ] · J [ l ] : k = l = k , l ∈ N are both even with k + l ≥ } . In order to show that B ( G ) is seminormal, let x ∈ q (cid:0) B ( G ) (cid:1) be given suchthat x [ ] , x [ ] ∈ B ( G ) . We have to show that x ∈ B ( G ) . Since x [ ] , x [ ] ∈ B ( G ) ⊂ F ( G ) and F ( G ) is seminormal, it follows that x ∈ F ( G ) . If x = I [ k ] with k ∈ N ,then I [ k ] ∈ B ( G ) implies that 4 | k , hence 4 | k , and thus x ∈ B ( G ) . Similarly, if x = J [ l ] ∈ B ( G ) with l ∈ N , then x ∈ B ( G ) . It remains to consider the case x = I [ k ] · J [ l ] with k , l ∈ N . Since x [ ] = I [ k ] · J [ l ] ∈ B ( G ) , it follows that k , l areboth even, and thus x ∈ B ( G ) . Therefore B ( G ) is seminormal. Transfer Homomorphisms
A well-established strategy for investigating the arithmetic of a given monoid H is to construct a transfer homomorphism q : H → B , where B is a simpler monoidthan H and the transfer homomorphism q allows to shift arithmetical results from B back to the (original, more complicated) monoid H . We will use transfer homomor-phisms in Section 4 in order to show that properties of the monoid of G -invariantmonomials can be studied in a monoid of zero-sum sequences (see Propositions 4.7and 4.9). Definition 3.5.
A monoid homomorphism q : H → B is called a transfer homo-morphism if it has the following properties: (T 1) B = q ( H ) B × and q − ( B × ) = H × . (T 2) If u ∈ H , b , c ∈ B and q ( u ) = bc , then there exist v , w ∈ H such that u = vw , q ( v ) B × = bB × and q ( w ) B × = cB × .We will use the simple fact that, if q : H → B and q ′ : B → B ′ are transfer homo-morphisms, then their composition q ′ ◦ q : H → B ′ is a transfer homomorphism too.The next proposition summarizes key properties of transfer homomorphisms. Proposition 3.6.
Let q : H → B be a transfer homomorphism and a ∈ H.1. a is an atom of H if and only if q ( a ) is an atom of B.2. L H ( a ) = L B (cid:0) q ( a ) (cid:1) , whence q (cid:0) M k ( H ) (cid:1) = M k ( B ) and q − (cid:0) M k ( B ) (cid:1) = M k ( H ) .3. If q is degree preserving, then D k ( H ) = D k ( B ) for all k ∈ N . Proof.
1. and 2. follow from [40, Proposition 3.2.3]. In order to prove 3., note thatfor all k ∈ N we have D k ( H ) = sup {| a | H : a ∈ M k ( H ) } = sup {| q ( a ) | B : q ( a ) ∈ M k ( B ) } = sup {| b | B : b ∈ M k ( B ) } = D k ( B ) . The first examples of transfer homomorphisms in the literature starts from aKrull monoid to its associated monoid of zero-sum sequences which is a Krullmonoid having a combinatorial flavor. These ideas were generalized widely, andthere are transfer homomorphisms from weakly Krull monoids to (simpler) weaklyKrull monoids (having a combinatorial flavor) and the same is true for C-monoids.
Proposition 3.7.
Let H be a Krull monoid, j : H → F ( P ) be a cofinal divisor ho-momorphism with class group G = C ( j ) , and let G ∗ ⊂ G denote the set of classescontaining prime divisors. Let e q : F ( P ) → F ( G ∗ ) denote the unique homomor-phism defined by e q ( p ) = [ p ] for all p ∈ P, and set q = e q ◦ j : H → B ( G ∗ ) .1. q is a transfer homomorphism.2. For a ∈ H, we set | a | = | j ( a ) | and for S ∈ B ( G ∗ ) we set | S | = | S | F ( G ∗ ) . Then | a | = | q ( a ) | for all a ∈ H, q ( M ∗ k ( H )) = M ∗ k ( G ∗ ) and q − ( M ∗ k ( G ∗ )) = M ∗ k ( H ) for all k ∈ N . Furthermore, e ( H ) = e ( G ∗ ) , h ( H ) = h ( G ∗ ) , and D k ( H ) = D k ( G ∗ ) for all k ∈ N .Proof.
1. follows from [40, Section 3.4]. By definition, we have | a | = | q ( a ) | for all a ∈ H . Thus the assertions on D k ( H ) follow from Proposition 2.7, and the remainingstatements can be derived in a similar way.The above transfer homomorphism q : H → B ( G ∗ ) constitutes the link betweenthe arithmetic of Krull monoids on the one side and zero-sum theory on the otherside. In this way methods from Arithmetic Combinatorics can be used to obtainprecise results for arithmetical invariants describing the arithmetic of H . For anoverview of this interplay see [37].There is a variety of transfer homomorphisms from monoids of zero-sum se-quences to monoids of zero-sum sequences in order to simplify specific structuralfeatures of the involved subsets of groups. Below we present a simple example ofsuch a transfer homomorphism which we will meet again in Proposition 4.9 (formore of this nature we refer to [74] and to [40, Theorem 6.7.11]). Let G be abelianand let G ⊂ G be a subset. For g ∈ G we define e ( G , g ) = gcd (cid:0) { v g ( B ) : B ∈ B ( G ) } (cid:1) , and it is easy to check that (for details see [45, Lemma 3.4]) e ( G , g ) = gcd (cid:0) { v g ( A ) : A ∈ A ( G ) } (cid:1) = min (cid:0) { v g ( A ) : v g ( A ) > , A ∈ A ( G ) } (cid:1) = min (cid:0) { v g ( B ) : v g ( B ) > , B ∈ B ( G ) } (cid:1) = min (cid:0) { k ∈ N : kg ∈ h G \ { g }i} (cid:1) = gcd (cid:0) { k ∈ N : kg ∈ h G \ { g }i} (cid:1) . nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 23 Proposition 3.8.
Let G be abelian and G , G , G ⊂ G be subsets such that G = G ⊎ G . For g ∈ G we set e ( g ) = e ( G , g ) and we define G ∗ = { e ( g ) g : g ∈ G } ∪ G . Then the map q : B ( G ) −→ B ( G ∗ ) B = (cid:213) g ∈ G g [ v g ( B )] (cid:213) g ∈ G ( e ( g ) g ) [ v g ( B ) / e ( g )] (cid:213) g ∈ G g [ v g ( B )] is a transfer homomorphism.Proof. Clearly, q is a surjective homomorphism satisfying q − ( F ( G ) ) = { F ( G ) } .In order to verify property (T2) of Definition 3.5, let B ∈ B ( G ) and C , C ∈ B ( G ∗ ) be such that q ( B ) = C · C . We have to show that there are B , B ∈ B ( G ) suchthat B = B · B , q ( B ) = C , and q ( B ) = C . This can be checked easily. The k th Davenport constants: the general case Let G ⊂ G be a subset, and k ∈ N . Recall that e ( G ) = max { ord ( g ) : g ∈ G } . If G is nilpotent, then G is the direct sum of its p -Sylow subgroups and hence e ( G ) = lcm { ord ( g ) : g ∈ G } = exp ( G ) . Let • E ( G ) be the smallest integer ℓ ∈ N such that every sequence S ∈ F ( G ) oflength | S | ≥ ℓ has a product-one subsequence of length | G | . • s ( G ) denote the smallest integer ℓ ∈ N such that every sequence S ∈ F ( G ) oflength | S | ≥ ℓ has a product-one subsequence of length e ( G ) .The Davenport constants, together with the Erd˝os-Ginzburg-Ziv constant s ( G ) ,the constants h ( G ) and E ( G ) , are the most classical zero-sum invariants whose study(in the abelian setting) goes back to the early 1960s. The k th Davenport constants D k ( G ) were introduced by Halter-Koch [50] and further studied in [40, Section 6.1]and [30] (all this work is done in the abelian setting). First results in the non-abeliansetting were achieved in [15].If G is abelian, then W. Gao proved that E ( G ) = | G | + d ( G ) . For cyclic groups thisis the Theorem of Erd˝os-Ginzburg-Ziv which dates back to 1961 ([40, Proposition5.7.9]). W. Gao and J. Zhuang conjectured that the above equality holds true for allfinite groups ([82, Conjecture 2]), and their conjecture has been verified in a varietyof special cases [3, 36, 34, 53]. For more in the non-abelian setting see [80, 79].We verify two simple properties occurring in the assumptions of Propositions 2.7and 2.8. • If S ∈ F ( G ) and g ∈ p ( S ) , then h = g − ∈ G is a prime in F ( G ) and h · S ∈ B ( G ) . • Clearly, 1 G ∈ B ( G ) is a prime element of B ( G ) . Therefore all properties proved in Propositions 2.7 and 2.8 for D k ( H ) and d k ( H ) hold for the constants D k ( G ) and d k ( G ) (the linearity properties as given in Proposi-tion 2.7.2 and Proposition 2.8.2.(b) were first proved by Freeze and W.A. Schmid incase of abelian groups G [30]). We continue with properties which are more specific. Proposition 3.9.
Let H ≤ G be a subgroup, N ⊳ G be a normal subgroup, and k , ℓ ∈ N .1. d k ( N ) + d ℓ ( G / N ) ≤ d k + ℓ − ( G ) .2. d k ( G ) ≤ d d k ( N )+ ( G / N ) .3. d k ( G ) + ≤ [ G : H ]( d k ( H ) + ) .4. d k ( G ) + ≤ k ( d ( G ) + ) .5. D k ( G ) ≤ [ G : H ] D k ( H ) .Proof.
1. Let S = ( g N ) · . . . · ( g s N ) ∈ M ∗ ℓ ( G / N ) with | S | = s = d ℓ ( G / N ) and let T = h · . . . · h t ∈ M ∗ k ( N ) with t = d k ( N ) . We consider the sequence W = g · . . . · g s · h · . . . · h t ∈ F ( G ) and suppose that it is divisible by S · . . . · S a · T · . . . · T b where S i , T j ∈ B ( G ) \ { F ( G ) } , supp ( S i ) ∩ { g , . . . , g s } 6 = /0 and T · . . . · T b | h · . . . · h t forall i ∈ [ , a ] and all j ∈ [ , b ] . For i ∈ [ , a ] , let S i ∈ F ( G / N ) denote the sequenceobtained from S i by replacing each g n by g n N and by omitting the elements of S i which lie in { h , . . . , h t } . Then S , . . . , S a ∈ B ( G / N ) \ { F ( G ) } and S · . . . · S a | S whence a ≤ ℓ −
1. By construction, we have b ≤ k − a + b < k + ℓ − W ∈ M ∗ k + ℓ − ( G ) , and | W | = s + t = d k ( N ) + d ℓ ( G / N ) ≤ d k + ℓ − ( G ) .2. We set m = d d k ( N )+ ( G / N ) +
1. By (1), we have to show that every sequence S over G of length | S | ≥ m is divisible by a product of k nontrivial product-one se-quences. Let f : G → G / N denote the canonical epimorphism and let S ∈ F ( G ) be asequence of length | S | ≥ m . By definition of m , there exist sequences S , . . . , S d k ( N )+ such that S · . . . · S d k ( N )+ | S and f ( S ) , . . . , f ( S d k ( N )+ ) are product-one sequencesover G / N . Thus, for each n ∈ [ , d k ( N ) + ] , there are elements h n ∈ N such that h n ∈ p ( S n ) . Then T = h · . . . · h d k ( N )+ is a sequence over N , and it has k nontrivialproduct-one subsequences T , . . . , T k whose product T · . . . · T k divides T . Thereforewe obtain k nontrivial product-one sequences whose product divides S .3. We set m = [ G : H ] and start with the following assertion. A. If S ∈ F ( G ) with | S | ≥ m , then P ( S ) ∩ H = /0. Proof of A . Let S = g · . . . · g n ∈ F ( G ) with | S | = n ≥ m . We consider theleft cosets g H , g g H , . . . , g . . . g m H . If one of these cosets equals H , then weare done. If this is not the case, then there are k , ℓ ∈ [ , m ] with k < ℓ such that g . . . g k H = g . . . g k g k + . . . g ℓ H which implies that g k + . . . g ℓ ∈ H . ⊓⊔ (Proof of A )Now let S ∈ F ( G ) be a sequence of length | S | = [ G : H ]( d k ( H ) + ) . We haveto show that S is divisible by a product of k nontrivial product-one sequences. By A , there are d k ( H ) + S , . . . , S d k ( H )+ and elements h , . . . , h d k ( H )+ ∈ H such that S · . . . · S d k ( H )+ | S and h n ∈ p ( S n ) for each n ∈ [ , d k ( H ) + ] . By defini-tion, the sequence h · . . . · h d k ( H )+ ∈ F ( H ) is divisible by a product of k nontrivial nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 25 product-one sequences. Therefore S is divisible by a product of k nontrivial product-one sequences.4. Let S ∈ F ( G ) be a sequence of length | S | = k ( d ( G ) + ) . Then S may bewritten as a product S = S · . . . · S k where S , . . . , S k ∈ F ( G ) with | S n | = d ( G ) + n ∈ [ , k ] . Then each S n is divisible by a nontrivial product-one sequence T n and hence S is divisible by T · . . . · T k . Thus by (1) we infer that d k ( G ) + ≤ k ( d ( G ) + ) .5. Let A = g · . . . · g ℓ ∈ B ( G ) with g . . . g ℓ = ℓ > [ G : H ] D k ( H ) . We showthat ℓ > D k ( G ) . We set d = D k ( H ) and consider the left H -cosets C j = g . . . g j H for each j ∈ [ , ℓ ] . By the pigeonhole principle there exist 1 ≤ i < · · · < i d + ≤ ℓ such that C i = · · · = C i d + . We set h s = g i s + . . . g i s + for each s ∈ [ , d ] and h d + = g i d + + . . . g ℓ g . . . g i − . Clearly h , . . . , h d + ∈ H , and g · · · g ℓ = h · · · h d + = h · . . . · h d + ∈ B ( H ) . The inequality d + > D k ( H ) im-plies that h · . . . · h d + = S · . . . · S k + , where 1 F ( H ) = S i ∈ B ( H ) for i ∈ [ , k + ] .Let T i ∈ F ( G ) denote the sequence obtained from S i by replacing each occurrenceof h s by g i s + · . . . · g i s + for s ∈ [ , d ] and h d + by g i d + + · . . . · g ℓ · g · . . . · g i − .Then T , . . . , T k + ∈ B ( G ) and A = g · . . . · g ℓ = T · . . . · T k + , which implies that ℓ > D k ( G ) .Much more is known for the classical Davenport constants D ( G ) = D ( G ) and d ( G ) = d ( G ) . We start with metacyclic groups of index two. The following resultwas proved in [39, Theorem 1.1]. Theorem 3.10.
Suppose that G has a cyclic, index subgroup. Then D ( G ) = d ( G ) + | G ′ | and d ( G ) = (cid:26) | G | − if G is cyclic | G | if G is non-cyclic,where G ′ = [ G , G ] is the commutator subgroup of G. The next result gathers upper bounds for the large Davenport constant (for d ( G ) see [35]). Theorem 3.11.
Let G ′ = [ G , G ] denote the commutator subgroup of G.1. D ( G ) ≤ d ( G ) + | G ′ | − , and equality holds if and only if G is abelian.2. If G is a non-abelian p-group, then D ( G ) ≤ p + p − p | G | .3. If G is non-abelian of order pq, where p , q are primes with p < q, then D ( G ) = qand d ( G ) = q + p − .4. If N ⊳ G is a normal subgroup with G / N ∼ = C p ⊕ C p for some prime p, then d ( G ) ≤ ( d ( N ) + ) p − ≤ p | G | + p − .
5. If G is non-cyclic and p is the smallest prime dividing | G | , then D ( G ) ≤ p | G | .6. If G is neither cyclic nor isomorphic to a dihedral group of order n with odd n,then D ( G ) ≤ | G | . Proof.
All results can be found in [48]: see Lemma 4.2, Theorems 3.1, 4.1, 5.1,7.1,7.2, and Corollary 5.7.
Corollary 3.12.
The following statements are equivalent : (a) G is cyclic or isomorphic to a dihedral group of order n for some odd n ≥ .(b) D ( G ) = | G | .Proof. If G is not as in (a), then D ( G ) ≤ | G | by Theorem 3.11.6. If G is cyclic, then D ( G ) = | G | by Lemma 3.1.3. If G is dihedral of order 2 n for some odd n ≥
3, thenthe commutator subgroup G ′ of G has order n and hence D ( G ) = | G | by Theorem3.10. The k th Davenport constants: The abelian case Throughout this subsection, all groups are abelian and will be written additively.
We have G ∼ = C n ⊕ . . . ⊕ C n r , with r ∈ N and 1 < n | . . . | n r , r ( G ) = r is the rank of G and n r = exp ( G ) is the exponent of G . We define d ∗ ( G ) = r (cid:229) i = ( n i − ) . If G = { } , then r = = d ∗ ( G ) . An s -tuple ( e , . . . , e s ) of elements of G \ { } is saidto be a basis of G if G = h e i ⊕ . . . ⊕ h e s i . First we provide a lower bound for theDavenport constants. Lemma 3.13.
Let G be abelian.1. D k ( G ) = + d k ( G ) for every k ∈ N .2. d ∗ ( G ) + ( k − ) exp ( G ) ≤ d k ( G ) .Proof.
1. Let k ∈ N . By Proposition 2.8.1, we have 1 + d k ( G ) ≤ D k ( G ) . Obviously,the map y : M ∗ k ( G ) → M k ( G ) \ { } , given by y ( S ) = ( − s ( S )) · S , is surjective and we have | y ( S ) | = + | S | for every S ∈ M ∗ k ( G ) . Therefore we have1 + d k ( G ) = D k ( G ) .2. Suppose that G ∼ = C n ⊕ . . . ⊕ C n r , with r ∈ N and 1 < n | . . . | n r . If ( e , . . . , e r ) is a basis of G with ord ( e i ) = n i for all i ∈ [ , r ] , then S = e [ n r ( k − )] r r (cid:213) i = e [ n i − ] i is not divisible by a product of k nontrivial zero-sum sequences whence d ∗ ( G ) +( k − ) exp ( G ) = | S | ≤ d k ( G ) . nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 27 We continue with a result on the k th Davenport constant which refines the moregeneral results in Subsection 2.5. It provides an explicit formula for d k ( G ) in termsof d ( G ) (see [40, Theorem 6.1.5]). Theorem 3.14.
Let G be abelian, exp ( G ) = n, and k ∈ N .1. Let H ≤ G be a subgroup such that G = H ⊕ C n . Then d ( H ) + kn − ≤ d k ( G ) ≤ ( k − ) n + max { d ( G ) , h ( G ) − n − } . In particular, if d ( G ) = d ( H ) + n − and h ( G ) ≤ d ( G ) + n + , then d k ( G ) = d ( G ) + ( k − ) n.2. If r ( G ) ≤ , then d k ( G ) = d ( G ) + ( k − ) n.3. If G is a p-group and D ( G ) ≤ n − , then d k ( G ) = d ( G ) + ( k − ) n. For the rest of this section we focus on the classical Davenport constant D ( G ) .By Lemma 3.13.2, there is the crucial inequality d ∗ ( G ) ≤ d ( G ) . We continue with a list of groups for which equality holds. The list is incompletebut the remaining groups for which d ∗ ( G ) = d ( G ) is known are of a similar specialnature as those listed in Theorem 3.15.3 (see [76] for a more detailed discussion).In particular, it is still open whether equality holds for all groups of rank three (see[76, Section 4.1]) or for all groups of the form G = C rn (see [47]). Theorem 3.15.
We have d ∗ ( G ) = d ( G ) in each of the following cases :
1. G is a p-group or has rank r ( G ) ≤ .2. G = K ⊕ C km where k , m ∈ N , p ∈ P a prime, m a power of p and K ≤ G is ap-subgroup with d ( K ) ≤ m − .3. G = C m ⊕ C mn where m ∈ { , , , } and n ∈ N .Proof. For 1. see [40] (in particular, Theorems 5.5.9 and 5.8.3) for proofs and his-torical comments. For 2. see [37, Corollary 4.2.13], and 3. can be found in [5] and[76, Theorem 4.1].There are infinite series of groups G with d ∗ ( G ) < d ( G ) . However, the true reasonfor the phenomenon d ∗ ( G ) < d ( G ) is not understood. Here is a simple observation.Suppose that G = C n ⊕ . . . ⊕ C n r with 1 < n | . . . | n r , I ⊂ [ , r ] , and let G ′ = ⊕ i ∈ I C n i .If d ∗ ( G ′ ) < d ( G ′ ) , then d ∗ ( G ) < d ( G ) . For series of groups G which have rank fourand five and satisfy d ∗ ( G ) < d ( G ) we refer to [44, 42]. A standing conjecture foran upper bound on D ( G ) states that d ( G ) ≤ d ∗ ( G ) + r ( G ) . However, the availableresults are much weaker ([40, Theorem 5.5.5], [6]).The remainder of this subsection is devoted to inverse problems with respect tothe Davenport constant. Thus the objective is to study the structure of zero-sum freesequences S whose lengths | S | are close to the maximal possible value d ( G ) . If G is cyclic of order n ≥
2, then an easy exercise shows that S is zero-sum freeof length | S | = d ( G ) if and only if S = g [ n − ] for some g ∈ G with ord ( g ) = n . Aftermany contributions since the 1980s, S. Savchev and F. Chen could finally prove a(sharp) structural result. In order to formulate it we need some more terminology. If g ∈ G is a nonzero element of order ord ( g ) = n and S = ( n g ) · . . . · ( n ℓ g ) , where ℓ ∈ N and n , . . . , n ℓ ∈ [ , n ] , we define k S k g = n + . . . + n ℓ n . Obviously, S has sum zero if and only if k S k g ∈ N , and the index of S is defined asind ( S ) = min {k S k g : g ∈ G with G = h g i} ∈ Q ≥ . Theorem 3.16.
Let G be cyclic of order | G | = n ≥ .1. If S is a zero-sum free sequence over G of length | S | ≥ ( n + ) / , then there existg ∈ G with ord ( g ) = n and integers = m , . . . , m | S | ∈ [ , n − ] such that • S = ( m g ) · . . . · ( m | S | g ) • m + . . . + m | S | < n and S ( S ) = { n g : n ∈ [ , m + . . . + m | S | ] } .2. If U ∈ A ( G ) has length | U | ≥ ⌊ n ⌋ + , then ind ( U ) = .Proof.
1. See [71] for the original paper. For the history of the problem and a proofin the present terminology see [37, Chapter 5.1] or [49, Chapter 11].2. This is a simple consequence of the first part (see [37, Theorem 5.1.8]).The above result was generalized to groups of the form G = C ⊕ C n by S.Savchev and F. Chen ([72]). Not much is known about the number of all minimalzero-sum sequences of a given group. However, the above result allows to give aformula for the number of minimal zero-sum sequences of length ℓ ≥ ⌊ n ⌋ + ℓ > n / Corollary 3.17.
Let G be cyclic of order | G | = n ≥ , and let ℓ ∈ h ⌊ n ⌋ + , n i . Thenthe number of minimal zero-sum sequences U ∈ A ( G ) of length ℓ equals F ( n ) p ℓ ( n ) ,where F ( n ) = | ( Z / n Z ) × | is Euler’s Phi function and p ℓ ( n ) is the number of integerpartitions of n into ℓ summands.Proof. Clearly, every generating element g ∈ G and every integer partition n = m + . . . + m ℓ gives rise to a minimal zero-sum sequence U = ( m g ) · . . . · ( m ℓ g ) .Conversely, if U ∈ A ( G ) of length | U | = ℓ , then Theorem 3.16.2 implies that thereis an element g ∈ G with ord ( g ) = n such that U = ( m g ) · . . . · ( m ℓ g ) where m , . . . , m ℓ ∈ [ , n − ] with n = m + . . . + m ℓ . ( ∗ )Since G has precisely F ( n ) generating elements, it remains to show that for every U ∈ A ( G ) of length | U | = ℓ there is precisely one generating element g ∈ G with nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 29 k U k g =
1. Let U be as in ( ∗ ) , and assume to the contrary that there are a ∈ [ , n − ] with gcd ( a , n ) = m ′ , . . . , m ′ ℓ ∈ [ , n ] such that m ′ + . . . + m ′ ℓ = n and U = (cid:0) m ′ ( ag ) (cid:1) · . . . · (cid:0) m ′ ℓ ( ag ) (cid:1) . Let a ′ ∈ [ , n − ] such that aa ′ ≡ ( mod n ) . Since n = m + . . . + m ℓ ≥ v g ( U ) + a v ag ( U ) + ( ℓ − v g ( U ) − v ag ( U ))= ℓ − v g ( U ) + ( a − ) v ag ( U ) and n = m ′ + . . . + m ′ ℓ ≥ a ′ v g ( U ) + v ag ( U ) + ( ℓ − v g ( U ) − v ag ( U ))= ℓ + ( a ′ − ) v g ( U ) − v ag ( U ) , it follows that ( a − ) n = n + ( a − ) n ≥ ℓ − v g ( U ) + ( a − ) v ag ( U ) + ( a − )( ℓ + ( a ′ − ) v g ( U ) − v ag ( U ))= ( a − ) ℓ + (( a − )( a ′ − ) − ) v g ( U ) , whence a = , a ′ = n + or a ′ = , a = n + because ℓ ≥ ⌊ n ⌋ +
2. By symmetry, wemay assume that a =
2. Then v g ( U ) ≥ ℓ − n ≥ ⌊ n ⌋ + − n ≥
3, and thus n ≥ a ′ v g ( U ) ≥ n + , a contradiction.The structure of all minimal zero-sum sequences of maximal length D ( G ) hasbeen completely determined for rank two groups ([31, 33, 75, 68]), for groups ofthe form G = C ⊕ C ⊕ C n with n ≥ G = C ⊕ C n with n ≥
70 ([8, Theorems 5.8 and 5.9]).
After gathering basic material from invariant theory in Subsection 4.1 we constructan explicit divisor theory for the algebra of polynomial invariants of a finite group(see Subsection 4.2). In Subsection 4.3 we present a detailed study of the abeliancase as outlined in the Introduction. In Subsection 4.4 we associate a BF-monoidto a G -module whose k th Davenport constant is a lower bound for the k th Noethernumber. Basics of invariant theory
Let n = dim F ( V ) and let r : G → GL ( n , F ) be a group homomorphism. Considerthe action of G on the polynomial ring F [ x , . . . , x n ] via F -algebra automorphisms in- duced by g · x j = (cid:229) ni = r ( g − ) ji x i . Taking a slightly more abstract point of departure,we suppose that V is a G -module (i.e. we suppose that V is endowed with an actionof G via linear transformations). Choosing a basis of V , V is identified with F n , thegroup GL ( n , F ) is identified with the group GL ( V ) of invertible linear transforma-tions of V , and F [ V ] = F [ x , . . . , x n ] can be thought of as the symmetric algebra of V ∗ , the dual G -module of V , in which ( x , . . . , x n ) is a basis dual to the standard ba-sis in V . The action on V ∗ is given by ( g · x )( v ) = x ( r ( g − ) v ) , where g ∈ G , x ∈ V ∗ , v ∈ V . Note that, if F is infinite, then F [ V ] is the algebra of polynomial functions V → F , and the action of G on F [ V ] is the usual action on functions V → F inducedby the action of G on V via r . Denote by F ( V ) the quotient field of F [ V ] , and extendthe G -action on F [ V ] to F ( V ) by g · f f = g · f g · f for f , f ∈ F [ V ] and g ∈ G . We define F ( V ) G = { f ∈ F ( V ) : g · f = f for all g ∈ G } ⊂ F ( V ) and F [ V ] G = F ( V ) G ∩ F [ V ] . Then F ( V ) G ⊂ F ( V ) is a subfield and F [ V ] G ⊂ F [ V ] is an F -subalgebra of F [ V ] ,called the ring of polynomial invariants of G (the group homomorphism r : G → GL ( V ) giving the G -action on V is usually suppressed from the notation). Sinceevery element of F ( V ) can be written in the form f f − with f ∈ F [ V ] and f ∈ F [ V ] G , it follows that F ( V ) G is the quotient field of F [ V ] G . Next we summarize somewell-known ring theoretical properties of F [ V ] G going back to E. Noether [64]. Theorem 4.1.
Let all notations be as above.1. F [ V ] G ⊂ F [ V ] is an integral ring extension and F [ V ] G is normal.2. F [ V ] is a finitely generated F [ V ] G -module, and F [ V ] G is a finitely generated F -algebra ( hence in particular a noetherian domain ) .3. F [ V ] G is a Krull domain with Krull dimension dim F ( V ) .Proof.
1. To show that F [ V ] G is normal, consider an element f ∈ F ( V ) G which isintegral over F [ V ] G . Then f is integral over F [ V ] as well, and since F [ V ] is normal,it follows that f ∈ F [ V ] ∩ F ( V ) G = F [ V ] G .To show that F [ V ] G ⊂ F [ V ] is an integral ring extension, consider an element f ∈ F [ V ] and the polynomial F f = (cid:213) g ∈ G ( X − g f ) ∈ F [ V ][ X ] . (2)The coefficients of F f are the elementary symmetric functions (up to sign) evaluatedat ( g f ) g ∈ G , and hence they are in F [ V ] G . Thus f is a root of a monic polynomial withcoefficients in F [ V ] G .2. For i ∈ [ , n ] , we consider the polynomials F x i ( X ) (cf. (2)), and denote by A ⊂ F [ V ] G ⊂ F [ V ] the F -algebra generated by the coefficients of F x , . . . , F x n . By nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 31 definition, A is a finitely generated F -algebra and hence a noetherian domain. Since x , . . . , x n are integral over A , F [ V ] = A [ x , . . . , x n ] is a finitely generated (and hencenoetherian) A -module. Therefore the A -submodule F [ V ] G is a finitely generated A -module, and hence a finitely generated F -algebra.3. By 1. and 2., F [ V ] G is an normal noetherian domain, and hence a Krull do-main by Theorem 2.1. Since F [ V ] G ⊂ F [ V ] is an integral ring extension, the Theo-rem of Cohen-Seidenberg implies that their Krull dimensions coincide, and hencedim ( F [ V ] G ) = dim ( F [ V ]) = dim F ( V ) .The algebra F [ V ] is graded in the standard way (namely, deg ( x ) = . . . = deg ( x n ) = F [ V ] G is generated by homogeneous elements. For F -subspaces S , T ⊂ F [ V ] we write ST for the F -subspace in F [ V ] spanned by all the products st ( s ∈ S , t ∈ T ) , and write S k = S . . . S (with k factors). The factor algebra of F [ V ] bythe ideal generated by F [ V ] G + is usually called the algebra of coinvariants . It inheritsthe grading of F [ V ] and is finite dimensional. Definition 4.2.
Let k ∈ N .1. Let b k ( G , V ) be the top degree of the factor space F [ V ] G + / ( F [ V ] G + ) k + , where F [ V ] G + is the maximal ideal of F [ V ] G spanned by the positive degree homoge-neous elements. We call b k ( G ) = sup { b k ( G , W ) : W is a G -module over F } the k th Noether number of G .2. Let b k ( G , V ) denote the top degree of the factor algebra F [ V ] / ( F [ V ] G + ) k F [ V ] andset b k ( G ) = sup { b k ( G , W ) : W is a G -module over F } . In the special case k = b ( G , V ) = b ( G , V ) , b ( G ) = b ( G ) , b ( G , V ) = b ( G , V ) , and b ( G ) = b ( G ) , and b ( G ) is the Noether number of G . If { f , . . . , f m } and { h , . . . , h l } are two min-imal homogeneous generating sets of F [ V ] G , then m = l and, after renumbering ifnecessary, deg ( f i ) = deg ( h i ) for all i ∈ [ , m ] ([61, Proposition 6.19]). Therefore bythe Graded Nakayama Lemma ([61, Proposition 8.31]) we have b ( G , V ) = max { deg ( f i ) : i ∈ [ , m ] } , where { f , . . . , f m } is a minimal homogeneous generating set of F [ V ] G . Again by theGraded Nakayama Lemma, b ( G , V ) is the maximal degree of a generator in a mini-mal system of homogeneous generators of F [ V ] as an F [ V ] G -module. If char ( F ) ∤ | G | ,then by [11, Corollary 3.2] we have b k ( G ) = b k ( G ) + b ( G , V ) ≤ b ( G , V ) + , (3)where the second inequality can be strict. If G is abelian, then b k ( G , V ) and b k ( G , V ) will be interpreted as k th Davenport constants (see Proposition 4.7). The regular G-module V reg has a basis { e g : g ∈ G } labelled by the group ele-ments, and the group action is given by g · e h = e gh for g , h ∈ G . More conceptually,one can identify V reg with the space of F -valued functions on G , on which G actslinearly via the action induced by the left multiplication action of G on itself. In thisinterpretation the basis element e g is the characteristic function of the set { g } ⊂ G .It was proved in [73] that, if char ( F ) =
0, then b ( G ) = b ( G , V reg ) . If F is alge-braically closed, each irreducible G -module occurs in V reg as a direct summand withmultiplicity equal to its dimension. Theorem 4.3.
1. If char ( F ) ∤ | G | , then b ( G ) ≤ | G | .2. If char ( F ) | | G | , then b ( G ) = ¥ .Proof.
1. The case char ( F ) = F is greater than | G | . The generalcase was shown independently by P. Fleischmann [25] and J. Fogarty [28] (see also[62, Theorem 2.3.3] and [57]. For 2. see [70].Bounding the Noether number has always been an objective of invariant theory(for recent surveys we refer to [81, 60]; degree bounds are discussed in [17, 78,26, 10, 54]; see [16] for algorithmic aspects). Moreover, the main motivation tointroduce the k th Noether numbers b k ( G ) ([11, 12, 13]) was to bound the ordinaryNoether number b ( G ) via structural reduction (see Subsection 5.1). The divisor theory of invariant rings
Let G ⊂ GL ( V ) and c ∈ Hom ( G , F • ) . Then F [ V ] G , c = { f ∈ F [ V ] : g · f = c ( g ) f for all g ∈ G } denotes the space of relative invariants of weight c , and we set F [ V ] G , rel = [ c ∈ Hom ( G , F • ) F [ V ] G , c . Clearly, we have F [ V ] G ⊂ F [ V ] G , rel ⊂ F [ V ] , and to simplify notation, we set H = ( F [ V ] G \ { } ) red , D = ( F [ V ] G , rel \ { } ) red , and E = ( F [ V ] \ { } ) red . Since F [ V ] is a factorial domain with F • as its set of units, E = F ( P ) is the freeabelian monoid generated by P = { F • f : f ∈ F [ V ] is irreducible } . The action of G on F [ V ] is via F -algebra automorphisms, so it induces a permutation action of G on E and P . Denote by P / G the set of G -orbits in P . We shall identify P / G with nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 33 a subset of E as follows: assign to the G -orbit { f , . . . , f r } the element f . . . f r ∈ E (here f , . . . , f r ∈ P are distinct).We say that a non-identity element g ∈ G ⊂ GL ( V ) is a pseudoreflection if ahyperplane in V is fixed pointwise by g , and g is not unipotent (this latter condi-tion holds automatically if char ( F ) does not divide | G | , since then a non-identityunipotent transformation cannot have finite order). We denote by Hom ( G , F • ) ≤ Hom ( G , F • ) the subgroup of the character group consisting of the characters thatcontain all pseudoreflections in their kernels. For each p ∈ P , choose a representa-tive ˜ p ∈ F [ V ] in the associate class p = F • ˜ p . We have X ( F [ V ]) = { ˜ p F [ V ] : p ∈ P } because F [ V ] is factorial. We set v ˜ p = v p : q ( F [ V ] • ) = F ( V ) • → Z , and for a subset X ⊂ F ( V ) we write v p ( X ) = inf { v p ( f ) : f ∈ X \ { }} . The ramification index of theprime ideal ˜ p F [ V ] over F [ V ] G is e ( p ) = v p ( ˜ p F [ V ] ∩ F [ V ] G ) . The ramification index e ( p ) can be expressed in terms of the inertia subgroupI p = { g ∈ G : g · f − f ∈ ˜ p F [ V ] for all f ∈ F [ V ] } . Since V ⋆ is a G -stable subspace in F [ V ] , the inertia subgroup I p acts trivially on V ⋆ / ( V ⋆ ∩ ˜ p F [ V ]) . On the other hand I p acts faithfully on V ⋆ . So if I p is non-trivial,then V ⋆ ∩ ˜ p F [ V ] =
0, implying ˜ p ∈ V ⋆ . Clearly I p must act trivially on the hyper-plane V ( ˜ p ) = { v ∈ V : ˜ p ( v ) = } , and hence acts via multiplication by a charac-ter d p ∈ Hom ( I p , F • ) on the 1-dimensional factor space V / V ( ˜ p ) . So ker ( d p ) isa normal subgroup of I p (necessarily unipotent hence trivial if char ( F ) ∤ | G | ) and I p = ker ( d p ) Z decomposes as a semi-direct product of ker ( d p ) and a cyclic sub-group Z consisting of pseudoreflections fixing pointwise V ( ˜ p ) . So Z ∼ = I p / ker ( d p ) is isomorphic to a finite subgroup of F • .The next Lemma 4.4 is extracted from Nakajima’s paper [58]. Lemma 4.4.
1. We have the equality e ( p ) = | Z | .2. v p ( F [ V ] G , c ) < e ( p ) for all c ∈ Hom ( G , F • ) .3. v p ( F [ V ] G , c ) = for all c ∈ Hom ( G , F • ) .Proof.
1. By [59, 9.6, Proposition (i)], we have that e ( p ) = v p ( ˜ p F [ V ] ∩ F [ V ] I p ) , theramification index of the prime ideal ˜ p F [ V ] over the subring of I p -invariants. Thusif I p is trivial, then e ( p ) =
1, and of course | Z | =
1. If I p is non-trivial, then as itwas explained above, ˜ p is a linear form, which is a relative I p -invariant with weight d − p , hence ˜ p | Z | is an I p -invariant, implying v p ( ˜ p F [ V ] ∩ F [ V ] I p ) ≤ | Z | . On the otherhand F [ V ] I p is contained in F [ V ] Z , and the algebra of invariants of the cyclic group Z fixing pointwise the hyperplane V ( ˜ p ) is generated by ˜ p | Z | and a subspace of V ∗ complementary to F ˜ p . Thus v p ( ˜ p F [ V ] ∩ F [ V ] I p ) ≥ v p ( ˜ p F [ V ] ∩ F [ V ] Z ) = | Z | , imply-ing in turn that e ( p ) = | Z | .2. Take an h ∈ F [ V ] G with e ( p ) = v p ( h ) . Note that v q ( h ) = v p ( h ) and v q ( F [ V ] G , c ) = v p ( F [ V ] G , c ) holds for all q ∈ G · p , since F [ V ] G , c is a G -stable subset in F [ V ] . Set S = { ft : f ∈ F [ V ] , t ∈ F [ V ] G \ ˜ p F [ V ] } . This is a G -stable subring in q ( F [ V ]) con-taining F [ V ] . Consider S c = S ∩ q ( F [ V ]) c , where q ( F [ V ]) c = { s ∈ q ( F [ V ]) : g · s = c ( g ) s for all g ∈ G } . Then v q ( S c ) = v q ( F [ V ] G , c ) for all q ∈ G · p , since for anydenominator t of an element ft of S we have v q ( t ) =
0. Now suppose that contraryto our statement we have e ( p ) ≤ v p ( F [ V ] G , c ) , and hence v q ( h ) ≤ v q ( S c ) for all q ∈ G · p . In particular this means that F [ V ] G , c = { } . Then v q ( h − S c ) ≥ q ∈ G · p . Now S is a Krull domain with X ( S ) = { ˜ qS : q ∈ G · p } , thus h − S c ⊂ S (see the discussion after Theorem 2.1), implying that S c ⊂ hS . Clearly hS ∩ S c = hS c ,so we conclude in turn that S c ⊂ hS c . Iterating this we deduce { } 6 = S c ⊂ ∩ ¥ n = h n S ,a contradiction.3. It is well known that F [ V ] G , c = { } (see the proof of A4. below). Write v = v p ( F [ V ] G , c ) . Take f ∈ F [ V ] G , c with v p ( f ) = v , say f = ˜ p v h , where h ∈ F [ V ] .Note that both f and ˜ p are relative invariants of I p , hence so is h . Therefore g · h ∈ F • h , and ˜ p | F [ V ] ( g · h − h ) for all g ∈ I p , implying that h is an I p -invariant. Any c ∈ Hom ( G , F • ) contains I p in its kernel (the unipotent normal subgroup ker ( d p ) of I p has no non-trivial characters at all, and Z = I p / ker ( d p ) consists of pseudore-flections). Thus f is I p -invariant as well. Therefore ˜ p v is I p -invariant, so its weight d vp is trivial. Consequently the order | Z | of d p in Hom ( I p , F • ) divides v . We have e ( p ) = | Z | by 1., and on the other hand v < e ( p ) by 2., forcing v = f , we denote by w ( f ) the weight of f . This inducesa homomorphism w : D → Hom ( G , F • ) assigning to F • f ∈ D the weight w ( f ) ofthe relative invariant f . Clearly, w extends to a group homomorphism w : q ( D ) → Hom ( G , F • ) . The kernel of w consists of elements of the form ( F • h ) − F • f , where f , h ∈ F [ V ] G , c for some character c . Now f / h belongs to F ( V ) G , which is the fieldof fractions of F [ V ] G , so there exist f , h ∈ F [ V ] G with f / h = f / h , implying ( F • h ) − F • f = ( F • h ) − F • f ∈ q ( H ) . Thus ker ( w ) = q ( H ) . Therefore w induces amonomorphism w : q ( D ) / q ( H ) → Hom ( G , F • ) . Theorem 4.5.
Let G ⊂ GL ( V ) , H = ( F [ V ] G \ { } ) red , and D = ( F [ V ] G , rel \ { } ) red .1. The embeddings F [ V ] G \ { } j ֒ → F [ V ] G , rel \ { } y ֒ → F [ V ] • are cofinal divisorhomomorphisms.2. D is factorial, P / G ⊂ E is the set of prime elements in D, and C ( j ) is a torsiongroup.3. The monoid D = { gcd D ( X ) : X ⊂ H finite } ⊂
D is free abelian with basis { q e ( q ) : q ∈ P / G } , where e ( q ) = min { v q ( h ) : q | D h ∈ H } , and the embeddingH ֒ → D is a divisor theory.4. We have D = { f ∈ D : w ( f ) ∈ Hom ( G , F • ) } and w | q ( D ) / q ( H ) : C ( F [ V ] G ) = q ( D ) / q ( H ) → Hom ( G , F • ) is an isomorphism. Theorem 4.5 immediately implies the following corollary which can be found inBenson’s book ([4, Theorem 3.9.2]) and which goes back to Nakajima [58] (see also[27] for a discussion of this theorem).
Corollary 4.6 (Benson-Nakajima).
The class group of F [ V ] G is isomorphic to Hom ( G , F • ) , the subgroup of the character group consisting of the characters thatcontain all pseudoreflections in their kernels. nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 35 Proof (of Theorem 4.5).
1. Since F [ V ] G = F ( V ) G ∩ F [ V ] , the embedding y ◦ j : F [ V ] G ֒ → F [ V ] is a divisor homomorphism, and hence j is a divisor homomorphism. Further-more, if the quotient of two relative invariants lies in F [ V ] , then it is a relative invari-ant whence y is a divisor homomorphism. In order to show that the embeddings arecofinal, let 0 = f ∈ F [ V ] be given. Then f ∗ = (cid:213) g ∈ G g f ∈ F [ V ] G and f | f ∗ , so theembedding y ◦ j is cofinal and hence j and y are cofinal.2. Suppose that { f , . . . , f r } ⊂ F [ V ] represents a G -orbit in P . Then g · ( f . . . f r ) is a non-zero scalar multiple of f . . . f r , hence f . . . f r ∈ F [ V ] G , rel . This shows that P / G ⊂ E is in fact contained in D . Conversely, take an irreducible element F • f in themonoid D (so f is a relative invariant). Take any irreducible divisor f of f in F [ V ] .Since g · f ∈ F • f , the polynomial g · f is also the divisor of f . Denoting by f , . . . , f r polynomials representing the G -orbit of F • f in P , we conclude that f . . . f r divides f in F [ V ] , hence F • f . . . f r divides F • f in D as well, so F • f . . . f r = F • f . Thisimplies that D is the submonoid of E = F ( P ) generated by P / G .In order to show that C ( j ) is a torsion group, let f ∈ D be given. We have to findan m ∈ N such that f m ∈ H . Clearly, this holds with m being the order in Hom ( G , F • ) of the weight of the relative invariant corresponding to f .3. Since C ( j ) is a torsion group, Proposition 2.2 implies that the embedding H ֒ → D is a divisor theory, and that D is free abelian with basis { q e ( q ) : q ∈ P / G } ,where e ( q ) = min { v q ( h ) : q | D h ∈ H } (note that if q ∈ P / G is the G -orbit of p ∈ P ,then v q ( h ) = v p ( h ) , where the latter is the exponent of p in h ∈ E = F ( P ) ).4. It remains to prove the following three assertions. A1. D = { f ∈ D : w ( f ) ∈ Hom ( G , F • ) } . A2. w ( D ) = Hom ( G , F • ) . A3. w | q ( D ) / q ( H ) : q ( D ) / q ( H ) → w ( D ) is an isomorphism. Proof of A1 . Set D = { f ∈ D : w ( f ) ∈ Hom ( G , F • ) } . We show first D ⊂ D .Let c be a character of G , and assume that c ( g ) = g ∈ G . Let f be a relative invariant with w ( f ) = c . Then for any v with gv = v wehave f ( v ) = f ( g − v ) = ( g f )( v ) = c ( g ) f ( v ) , hence f ( v ) =
0. So l | F [ V ] f , where l is a non-zero linear form on V that vanishes on the reflecting hyperplane of g . De-noting by l = l , . . . , l r representatives of the G -orbit of F • l , we find that the relativeinvariant q = l . . . l r divides f . Thus gcd D { f ∈ D | w ( f ) = c } 6 =
1. Now suppose thatfor some F • k ∈ D we have that w ( k ) does not belong to Hom ( G , F • ) . By defini-tion of D there exist h , . . . , h n ∈ D with gcd D ( h , . . . , h n ) = kh , . . . , kh n ∈ H . Clearly w ( h i ) = w ( k ) − / ∈ Hom ( G , F • ) , hence by the above considerationsgcd D ( h , . . . , h n ) =
1, a contradiction.Next we show D ⊂ D . Let d be an element in the monoid D . By Lemma 4.4.3for any prime divisor p ∈ P of d there exists an h p ∈ D such that w ( h p ) = w ( d ) − and p ∤ E h p . Denote by m > w ( d ) in the group of characters. Clearly d m ∈ H and dh p ∈ H . Moreover, gcd E ( d m , dh p : p ∈ P , p | E d ) = d . Proof of A2 . The statement follows from A1 , as soon as we show that F [ V ] G , c = c ∈ Hom ( G , F • ) . For any character c ∈ Hom ( G , F • ) the group ¯ G = G / ker ( c ) is isomorphic to a cyclic subgroup of F • , hence its order is not divisible by char ( F ) . Moreover, ¯ G acts faithfully on the field T = F ( V ) ker ( c ) , with T ¯ G = F ( V ) G .By the Normal Basis Theorem, T as a ¯ G -module over T ¯ G is isomorphic to the regu-lar representation of ¯ G , hence contains the representation c as a summand with mul-tiplicity 1. This shows in particular that T ¯ G contains a relative invariant of weight c .Multiplying this by an appropriate element of T ¯ G ∩ F [ V ] = F [ V ] G we get an elementof F [ V ] G , c . So all characters of G occur as the weight of a relative invariant in F [ V ] . Proof of A3 . Since w : q ( D ) / q ( H ) → Hom ( G , F • ) is a monomorphism, themap w | q ( D ) / q ( H ) : q ( D ) / q ( H ) → w ( q ( D )) is an isomorphism. Note finally that w ( q ( D )) = q ( w ( D )) = w ( D ) .As already mentioned, not only the class group but also the distribution of primedivisors in the classes is crucial for the arithmetic of the domain. Moreover, the classgroup together with the distribution of prime divisors in the classes are characteristic(up to units) for the domain. For a precise formulation we need one more definition.Let H be a Krull monoid, H red ֒ → F ( P ) a divisor theory, and let G be an abeliangroup and ( m g ) g ∈ G be a family of cardinal numbers. We say that H has charac-teristic ( G , ( m g ) g ∈ G ) if there is a group isomorphism F : G → C ( H ) such that m g = | P ∩ F ( g ) | . Two reduced Krull monoids are isomorphic if and only if theyhave the same characteristic ([40, Theorem 2.5.4]). We pose the following problem. Problem 1.
Let G be a finite group, F be a field, and V be a finite dimensional F -vector space endowed with a linear action of G . Determine the characteristic of F [ V ] G .Let all assumptions be as in Problem 1 and suppose further that G acts triviallyon one variable. Then F [ V ] G is a polynomial ring in this variable and hence everyclass contains a prime divisor by [29, Theorem 14.3]. The abelian case
Throughout this subsection, suppose that G is abelian, F is algebraically closed,and char ( F ) ∤ | G | . The assumption on algebraic closedness is not too restrictive, since for any field F the set F [ V ] G spans the ring of invariants over the algebraic closure F as a vectorspace over F . The assumption on the characteristic guarantees that every G -moduleis completely reducible (i.e. is the direct sum of irreducible G -modules). The dualspace V ∗ has a basis { x , . . . , x n } consisting of G -eigenvectors whence g · x i = c i ( g ) x i for all i ∈ [ , n ] where c , . . . , c n ∈ Hom ( G , F • ) . We set b G = Hom ( G , F • ) , b G V = { c , . . . , c n } ⊂ b G , and note that G ∼ = b G . Recall that a completely reducible H -module W (for a not necessarily abelian group H ) is called multiplicity free if it is the directsum of pairwise non-isomorphic irreducible H -modules. In our case V is multiplicityfree if and only if the characters c , . . . , c n are pairwise distinct. nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 37 It was B. Schmid ([73, Section 2]) who first formulated a correspondence be-tween a minimal generating system of F [ V ] G and minimal product-one sequencesover the character group (see also [24]). The next proposition describes in detailthe structural interplay. In particular, Proposition 4.7.2 shows that all (direct andinverse) results on minimal zero-sum sequences over b G V (see Subsections 3.3 and3.4) carry over to A ( M G ) . Proposition 4.7.
Let M ⊂ F [ x , . . . , x n ] be the multiplicative monoid of monomials, y : M → F ( b G V ) be the unique monoid homomorphism defined by y ( x i ) = c i forall i ∈ [ , n ] , and let M G ⊂ M denote the submonoid of G-invariant monomials.1. F [ V ] G has M G as an F -vector space basis, and F [ V ] G is minimally generated asan F -algebra by A ( M G ) .2. The homomorphism y : M → F ( b G V ) and its restriction y | M G : M G → B ( b G V ) are degree-preserving transfer homomorphisms. Moreover, M G is a reducedfinitely generated Krull monoid, and A ( M G ) = y − (cid:0) A ( b G V ) (cid:1) .3. y | M G is an isomorphism if and only if V is a multiplicity free G-module.4. b k ( G , V ) = D k ( M G ) = D k ( b G V ) and b k ( G ) = D k ( G ) for all k ∈ N .Proof.
1. Each monomial spans a G -stable subspace in F [ V ] , hence a polynomial is G -invariant if and only if all its monomials are G -invariant, so M G spans F [ V ] G . Theelements of M G are linearly independent, therefore F [ V ] G can be identified with themonoid algebra of M G over F , which shows the second statement.2. M and F ( b G V ) are free abelian monoids and y maps primes onto primes. Thus y : M → F ( b G V ) is a surjective degree-preserving monoid homomorphism and it isa transfer homomorphism. Let p : F ( b G ) → b G be the monoid homomorphism de-fined by p ( c ) = c for all c ∈ b G . Then ker ( p ) = B ( b G ) . Taking into account that fora monomial m ∈ M G acts on the space F m via the character p ( y ( m )) , we concludethat for a monomial m ∈ M we have that m ∈ M G if and only if y ( m ) ∈ B ( b G V ) .This implies that the restriction y | M G of the transfer homomorphism y is also atransfer homomorphism. Therefore M G is generated by A ( M G ) = y − (cid:0) A ( b G V ) (cid:1) .Since A ( b G V ) is finite, and y has finite fibers, we conclude that the monoid M G isfinitely generated. Since M is factorial and F [ V ] G ⊂ F [ V ] is saturated by Theorem4.5, it follows that M ∩ q ( M G ) ⊂ M ∩ F [ V ] ∩ q ( F [ V ] G ) ⊂ M ∩ F [ V ] G = M G whence M G ⊂ M is saturated and thus M G is a Krull monoid.3. V is a multiplicity free G -module if and only if c , . . . , c n are pairwise distinct.Since y : M → F ( b G V ) maps the primes x , . . . , x n of M onto the primes c , . . . , c n of F ( b G V ) , y is an isomorphism if and only if c , . . . , c n are pairwise distinct.4. Let k ∈ N and M G + = M G \ { } . Then M G \ ( M G + ) k + = M k ( M G ) . Since y | M G : M G → B ( b G V ) is degree-preserving transfer homomorphism, Proposition 3.6.3implies that D k ( M G ) = D k ( b G V ) . Since F [ V ] G is spanned by M G , ( F [ V ] G + ) k + isspanned by ( M G + ) k + . Therefore the top degree of a homogeneous G -invariant not contained in ( F [ V ] G + ) k + coincides with the maximal degree of a monomial in M G + \ ( M G + ) k + = M k ( M G ) . Thus b k ( G , V ) = D k ( M G ) . For the k th Noether number b k ( G ) we have b k ( G ) = sup { b k ( G , W ) : W is a G -module over F } = sup { D k ( b G W ) : W is a G -module over F } = D k ( b G ) because for the regular representation V reg we have b G V reg = b G .Recalling the notation of Theorem 4.5, we have H = ( F [ V ] G \{ } ) red and D = { gcd D ( X ) : X ⊂ H finite } ⊂ D = ( F [ V ] G , rel \{ } ) red . Furthermore, M ⊂ F [ V ] = F [ x , . . . , x n ] is the monoid of monomials, M G = M ∩ F [ V ] G , and we can view M as a submonoid of H and then M G = M ∩ H . Since M ⊂ H is saturated, M = q ( M ) ∩ H , and q ( M ) / q ( M G ) = q ( M ) / q ( M ∩ H ) = q ( M ) / ( q ( M ) ∩ q ( H )) ∼ = q ( M ) q ( H ) / q ( H ) ⊂ q ( D ) / q ( H ) , we consider q ( M ) / q ( M G ) as a subset of q ( D ) / q ( H ) . Proposition 4.8.
Let all notation be as above and set M = M ∩ D .1. M ⊂ D is divisor closed whence M is free abelian, and A ( M ) = M ∩ A ( D ) = { x e ( x ) , . . . , x e ( x n ) n } .2. We have e ( x i ) = min { k ∈ N : c ki ∈ h c j | j = i i} .3. Hom ( r ( G ) , F • ) is generated by { c e ( x ) , . . . , c e ( x n ) n } and F [ x e ( x ) , . . . , x e ( x n ) n ] = F [ V ] G , where G denotes the subgroup of r ( G ) generated by the pseudoreflec-tions in r ( G ) .4. The embedding M G ֒ → M is a divisor theory,w | q ( M ) / q ( M G ) : C ( M G ) = q ( M ) / q ( M G ) → Hom ( r ( G ) , F • ) is an isomorphism, and w ( C ( M G ) ∗ ) = { c e ( x ) , . . . , c e ( x n ) n } .Proof.
1. If the product of two polynomials in F [ V ] has a single non-zero term, thenboth polynomials must have only one non-zero term. Thus, if ab ∈ M for some a , b ∈ D , then both a and b belong to M . Hence M ⊂ D is divisor closed implyingthat M ⊂ D is divisor-closed. Therefore A ( M ) = M ∩ A ( D ) .By Theorem 4.5.3, A ( D ) = { q e ( q ) : q ∈ A ( D ) } . The divisor closedness of M in D implies that if q e ( q ) ∈ M , then q ∈ M ∩ A ( D ) = A ( M ) = { x , . . . , x n } . Thus M ∩ A ( D ) = { x e ( x ) , . . . , x e ( x n ) n } .2. For i ∈ [ , n ] , we have e ( x i ) = min { v x i ( h ) : x i | D h , h ∈ H } = min { v x i ( m ) : x i | D m , m ∈ M G } , nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 39 where the second equality holds because for all h ∈ H we have v x i ( h ) = min { v x i ( m ) : m ranges over the monomials of h } . Note that a monomial m = (cid:213) ni = x a i i lies in M G if and only if (cid:213) ni = c [ a i ] i is a product-one sequence over b G if and only if c a i i = (cid:213) j = i c − a j j . Thus min { v x i ( m ) : x i | D m , m ∈ M G } = min { k ∈ N : c ki ∈ h c j | j = i i} .3. By Theorem 4.5.4, Hom ( r ( G ) , F • ) = w ( D ) and hence Hom ( r ( G ) , F • ) is generated by w ( A ( D )) . Thus by 1., it remains to show that h w ( A ( D )) i = h w ( A ( M )) i . Since A ( M ) ⊂ A ( D ) , it follows that h w ( A ( D )) i ⊃ h w ( A ( M )) i .To show the reverse inclusion, let a ∈ A ( D ) . For any monomial m occurring in a ,we have w ( m ) = w ( a ) . By Theorem 4.5.4, D = { f ∈ D : w ( f ) ∈ Hom ( r ( G ) , F • ) } whence m ∈ M ∩ D = M and clearly w ( m ) ∈ h w ( A ( M )) i .Recall that each monomial in F [ V ] spans a G -invariant subspace. Thus f ∈ F [ V ] is G -invariant if and only if all monomials of f are G -invariant. Furthermore, amonomial m is G -invariant if and only if w ( m ) contains G in its kernel; equiva-lently (by the characterization of D ) m ∈ M ∩ D = M . Thus F [ V ] G is generatedby A ( M ) and hence the assertion follows from 1.4. Since M ⊂ D , M ⊂ D and M G ⊂ H are divisor closed and since the embed-ding H ⊂ D is a divisor theory (Theorem 4.5.4), M G ֒ → M is a divisor homomor-phism into a free abelian monoid. Let m ∈ M . Then m ∈ D and there is a finitesubset Y ⊂ H such that m = gcd D ( Y ) . Let X ⊂ D ∩ M = M be the set of all mono-mials occurring in some y ∈ Y . Then m = gcd D ( X ) = gcd M ( X ) , where the lastequality holds because M ⊂ D is divisor closed.Restricting the isomorphism w | q ( D ) / q ( H ) : C ( F [ V ] G ) = q ( D ) / q ( H ) → Hom ( r ( G ) , F • ) from Theorem 4.5, we obtain a monomorphism w | q ( M ) / q ( M G ) : C ( M G ) = q ( M ) / q ( M G ) → Hom ( r ( G ) , F • ) . By 1. and 3., the image contains the generating set { c e ( x ) , . . . , c e ( x n ) n } of the groupHom ( r ( G ) , F • ) and hence the above monomorphism is an isomorphism. The laststatement follows from 1. by w ( C ( M G ) ∗ ) = w ( A ( M )) . Proposition 4.9.
Let M ⊂ F [ x , . . . , x n ] be the multiplicative monoid of monomials,and M G ⊂ M the submonoid of G-invariant monomials.1. Every class of C ( F [ V ] G ) contains a prime divisor.2. We have the following commutative diagram of monoid homomorphisms H q / / B ( C ( H )) w ∼ = / / B ( Hom ( r ( G ) , F • )) B ( b G V ) n ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ M G q / / y | MG ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ?(cid:31) O O B ( C ( M G ) ∗ ) ?(cid:31) w O O where • q and q are transfer homomorphisms of Krull monoids as given in Proposi-tion 3.7. • w is the extension to the monoid of product-one sequences of the group iso-morphism w | q ( D ) / q ( H ) given in Theorem 4.5.4 • w is the extension to the monoid of product-one sequences of the restrictionto C ( M G ) ∗ of the group isomorphism w | q ( M ) / q ( M G ) given in Proposition 4.8 • y is given in Proposition 4.7. • n will be defined below (indeed, n is a transfer homomorphism as given inProposition 3.8).3. If b G V = b G, then every class of C ( M G ) contains a prime divisor.Proof.
1. By Proposition 4.7.1, F [ V ] G is the monoid algebra of M G over F . Thus, by[7, Theorem 8], every class of F [ V ] G contains a prime divisor.2. In order to show that the diagram is commutative, we fix an m ∈ M G . Weconsider the divisor theory M G ֒ → M from Proposition 4.8 and factorize m in M ,say m = (cid:213) ni = (cid:0) x e ( x i ) i (cid:1) a i where a , . . . , a n ∈ N . Since w ( x e ( x i ) i ) = c e ( x i ) i for all i ∈ [ , n ] , it follows that ( w ◦ q )( m ) = ( c e ( x ) ) [ a ] · . . . · ( c e ( x n ) n ) [ a n ] ∈ B ( Hom ( r ( G ) , F • )) . Next we view m as an element in H and consider the divisor theory H ֒ → D . Since M ⊂ D is divisor closed, m = (cid:213) ni = (cid:0) x e ( x i ) i (cid:1) a i is a factorization of m in D . Therefore ( w ◦ q )( m ) = ( w ◦ q )( m ) .By definition of y , we infer that y ( m ) = c [ e ( x ) a ] · . . . · c [ e ( x n ) a n ] n . We define a partition of b G V = G ∪ G , where G = { c i : c i = c j for some distinct i , j ∈ [ , n ] } and G = b G V \ G . Let n : B ( b G V ) → B ( Hom ( r ( G ) , F • )) be defined as inProposition 3.8 (with respect to the partition G = G ⊎ G ). By Proposition 4.8.2, e ( x i ) = c i ∈ G , and e ( x i ) equals the number e ( c i ) in Proposition 3.8 if c i ∈ G .Therefore it follows that n ( y ( m )) = ( c e ( x ) ) [ a ] · . . . · ( c e ( x n ) n ) [ a n ] , nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 41 and hence the diagram commutes.3. In a finite abelian group all elements are contained in the subgroup generatedby the other elements, with the only exception of the generator of a 2-element group.Therefore unless G is the 2-element group and the non-trivial character occurs withmultiplicity one in the sequence c · . . . · c n , all the e ( x i ) = x i are all prime in M , so they represent all the divisor classes,as i varies in [ , n ] . In the missing case we have F [ V ] G = F [ x , . . . , x n − , x n ] (after arenumbering of the variables if necessary), hence both class groups are trivial, and x and x are prime elements in the unique class.Thus Proposition 4.9.1 gives a partial answer to Problem 1. Using that notationit states that m g ≥ g ∈ C ( F [ V ] G ) . Example 4.10.
The set C ( M G ) ∗ may be a proper subset of C ( M G ) , and conse-quently the monoid homomorphism n : B ( b G V ) → B ( Hom ( r ( G ) , F • )) is not sur-jective in general.1. Indeed, let G be cyclic of order 3, g ∈ G with ord ( g ) =
3, and the actionon F [ x , x , x ] is given by g · x i = w x i , where w is a primitive cubic root of 1. Then c = c = c = c , so e ( x ) = e ( x ) = e ( x ) =
1, implying w ( C ( M G ) ∗ ) = { c } (eachof the x i is a prime element in the class c ), whereas w ( C ( M G )) = { c , c , c = } ,the 3-element group. Thus B ( b G V ) = { c [ k ] : k ∈ N } , and n ( B ( b G V )) is the freeabelian monoid F ( { c } ) generated by c = ∈ b G . The polynomials x + x x and x + x x are irreducible, they are relative invariants of weight c and c , so theyrepresent prime elements of D in the remaining classes c and c = G be cyclicof order 5, g ∈ G with ord ( g ) =
5, and the action on F [ x , x , x ] is given by g · x = w x , g · x = w x , g · x = w x , where w is a primitive fifth root of 1.Then setting c = c , we have c = c , c = c and w ( C ( M G )) = h c i is the 5-element group, so V is multiplicity free. Still we have e ( x ) = e ( x ) = e ( x ) = w ( C ( M G ) ∗ ) = { c , c , c } (and x , x , x are the prime elements of M in theseclasses). The remaining classes c and c = D represented by x + x x and x + x x . A monoid associated with G -modules Throughout this subsection, suppose that char ( F ) ∤ | G | . In this subsection we discuss a monoid associated with representations of notnecessarily abelian groups which in the case of abelian groups recovers the monoidof G -invariant monomials. Decompose V into the direct sum of G -modules: V = V ⊕ ... ⊕ V r (4) and denote by r i : G → GL ( V i ) the corresponding group homomorphisms. Then (4)induces a decomposition of F [ V ] into multihomogeneous components as follows.The coordinate ring F [ V ] is the symmetric algebra Sym ( V ∗ ) = L ¥ n = Sym n ( V ∗ ) .Writing F [ V ] a = Sym a ( V ∗ ) ⊗ ... ⊗ Sym a r ( V ∗ r ) we have Sym n ( V ∗ ) = ⊕ | a | = n F [ V ] a ,and hence F [ V ] = ⊕ a ∈ N r F [ V ] a . The summands F [ V ] a are G -submodules in F [ V ] , and F [ V ] a F [ V ] b ⊂ F [ V ] a + b , so F [ V ] is a N r -graded algebra. Moreover, F [ V ] G is spannedby its multihomogeneous components F [ V ] Ga = F [ V ] G ∩ F [ V ] a . For f ∈ F [ V ] a wecall a the multidegree of f . We are in the position to define B ( G , V ) = { a ∈ N r : F [ V ] Ga = { }} (5)the set of multidegrees of multihomogeneous G -invariants. We give precise infor-mation on B ( G , V ) in terms of quantities associated to the direct summands V i of V .For i ∈ [ , r ] denote by c i the greatest common divisor of the elements of B ( G , V i ) ,and F i the Frobenius number of the numerical semigroup B ( G , V i ) ⊂ N , so F i isthe minimal positive integer N such that B ( G , V i ) contains N + kc i for all k ∈ N . Proposition 4.11. B ( G , V ) ⊂ N r is a reduced finitely generated C -monoid.2. For each i ∈ [ , r ] and all a ∈ N r satisfying a i ≥ b ( G , V i ) + F i we havea ∈ B ( G , V ) if and only if c i e i + a ∈ B ( G , V ) . (6)
3. For each i ∈ [ , r ] we have c i = | r i ( G ) ∩ F • id V i | .Proof.
1. Take a , b ∈ B ( G , V ) , so there exist non-zero f ∈ F [ V ] Ga and h ∈ F [ V ] Gb .Now 0 = f h ∈ F [ V ] Ga + b , hence a + b ∈ B ( G , V ) . This shows that B ( G , V ) is a sub-monoid of N . Moreover, the multidegrees of a multihomogeneous F -algebra gen-erating system of F [ V ] G clearly generate the monoid B ( G , V ) . Thus B ( G , V ) isfinitely generated by Theorem 4.1.To show that B ( G , V ) is also a C-monoid, recall that by Proposition 2.6.3 afinitely generated submonoid H of N r is a C-monoid if and only if each standardbasis element e i ∈ N r has a multiple in H . Now this condition holds for B ( G , V ) ,since by Theorem 4.1.2 F [ V i ] G ⊂ F [ V ] G contains a homogeneous element of positivedegree for each i ∈ [ , r ] .2. By symmetry it is sufficient to verify (6) in the case i =
1. Suppose a ∈ B ( G , V ) , so there is a non-zero G -invariant f ∈ Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r ) . De-compose Sym a ( V ∗ ) = L j W j into a direct sum of irreducible G -modules. Thisgives a direct sum decomposition Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r ) = L j ( W j ⊗ Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r )) . It follows that Sym a ( V ∗ ) contains an irreducible G -module di-rect summand W such that W ⊗ Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r ) contains a non-zero G -invariant. By definition of b ( G , V ) we know that F [ V ] is generated as an F [ V ] G module by its homogeneous components of degree ≤ b ( G , V ) . Therefore there ex-ists a d ≤ b ( G , V ) such that the degree d homogeneous component of F [ V ] con-tains a G -submodule U ∼ = W , and a ∈ d + B ( G , V ) . Now for any homogeneous nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 43 h ∈ F [ V ] G we have hU ⊗ Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r )) ⊂ F [ V ] ( d + deg ( h ) , a ,..., a r ) contains a non-zero G -invariant, since it is isomorphic to W ⊗ Sym a ( V ∗ ) ⊗ . . . ⊗ Sym a r ( V ∗ r )) . It follows that ( k , a , . . . , a r ) ∈ B ( G , V ) for all k ∈ d + B ( G , V ) , inparticular, for all k ∈ { d + F , d + F + c , d + F + c , . . . } .3. Let i ∈ [ , r ] , and to simplify notation set W = V i , c = c i , and f = r i . Recallthat F [ W ] A = F [ W ] B for some finite subgroups A , B ⊂ GL ( W ) implies that A = B .Indeed, the condition implies equality F ( W ) A = F ( W ) B of the corresponding quo-tient fields, and so both A and B are the Galois groups of the field extension F ( W ) over F ( W ) A = F ( W ) B , implying A = B . Now denote by Z ⊂ GL ( W ) the subgroupof scalar transformations Z = { w id W : w c = } , so Z is a central cyclic subgroupof GL ( W ) of order c . Clearly every homogeneous element of F [ W ] whose degreeis a multiple of c is invariant under Z . It follows that F [ W ] G ⊂ F [ W ] Z , hence de-noting by ˜ G the subgroup f ( G ) Z of GL ( W ) , we have F [ W ] G = F [ W ] ˜ G . It followsthat f ( G ) = ˜ G , i.e. Z ⊂ f ( G ) , and so c = | Z | divides the order of f ( G ) ∩ F • id W .Conversely, if l id W belongs to r ( G ) , then every element of F [ W ] G must be invari-ant under the scalar transformation l id W , whence all homogeneous components of F [ W ] G have degree divisible by the order of l , so the order of the cyclic group f ( G ) ∩ F • id W must divide c .In general B ( G , V ) is not a Krull monoid. To provide an example, consider thetwo-dimensional irreducible representation V of the symmetric group S = D . Itsring of polynomial invariants is generated by an element of degree 2 and 3, hence B ( G , V ) = h , i ⊂ ( N , +) , which is not Krull. Proposition 4.12.
For every k ∈ N we have D k ( B ( G , V )) ≤ b k ( G , V ) .Proof. Let k ∈ N . Take a ∈ B ( G , V ) such that | a | > b k ( G , V ) . By (5) a multi-homogeneous invariant f ∈ F [ V ] Ga exists. As deg ( f ) = | a | > b k ( G , V ) it followsthat f = (cid:229) Ni = f i , . . . f i , k + for some non-zero multihomogeneous invariants f i , j of positive degree. Denoting by a i , j ∈ N r the multidegree of f i , j , we have that a = a i , + . . . + a i , k + , where 0 = a i , j ∈ B ( G , V ) . This shows that all a ∈ B ( G , V ) with | a | > b k ( G , V ) factor into the product of more than k atoms, implying the de-sired inequality. Remarks.
1. Let G be abelian and suppose that F is algebraically closed. Thenwe may take in (4) a decomposition of V into the direct sum of 1-dimensional sub-modules and so V ∗ i , is spanned by a variable x i as in Subsection 4.3. Then F [ V ] a is spanned by the monomial x a · · · x a r r and a ∈ B ( G , V ) holds if and only if thecorresponding monomial is G -invariant. So in this case B ( G , V ) can be naturallyidentified with M G and the transfer homomorphism y | M G of Proposition 4.7 canbe thought of as a transfer homomorphism B ( G , V ) → B ( b G V ) , which is an iso-morphism if V is multiplicity free. However, this transfer homomorphism does notseem to have an analogues for non-abelian G (i.e. the study of B ( G , V ) can not bereduced to the multiplicity free case), as it is shown by the example below.2. The binary tetrahedral group G = e A ∼ = SL ( F ) of order 24 has a 2-dimensionalcomplex irreducible representation V such that F [ V ] G is minimally generated by elements of degree 6 , ,
12 (see for example [4, Appendix A]), hence B ( G , V ) = { , , , , , , , . . . } . On the other hand under this representation G is mappedinto the special linear group of V , so on V ⊕ V the function maping (( x , x ) , ( y , y )) det (cid:18) x y x y (cid:19) is a G -invariant of multidegree ( , ) , implying that ( , ) ∈ B ( G , V ⊕ V ) . This shows that the transfer homomorphism t : N → N , ( a , a ) a + a does not map B ( G , V ⊕ V ) into B ( G , V ) , as t ( , ) = / ∈ B ( G , V ) .Recall that the multigraded Hilbert series of F [ V ] G in r indeterminates T =( T , ..., T r ) is H ( F [ V ] G , T ) = (cid:229) a ∈ N r dim F ( F [ V ] Ga ) T a · · · T a r r , and hence B ( G , V ) = { a ∈ N r : the coefficient of T a in H ( F [ V ] G , T ) is nonzero } . By this observation Proposition 4.12 can be used for finding lower bounds on theNoether number b ( G , V ) , thanks to the following classical result of Molien (see forexample [4, Theorem 2.5.2]): Proposition 4.13.
Given a G-module V = V ⊕ ... ⊕ V r over C , let r i ( g ) ∈ GL ( V i ) be the linear transformation defining the action of g ∈ G on V i . Then we haveH ( C [ V ] G , T ) = | G | (cid:229) g ∈ G r (cid:213) i = ( id V i − r i ( g ) · T i ) . Example 4.14 (see p. 54-55 in [62]).
Consider the alternating group A and its 3-dimensional representation over C as the group of symmetries of an icosahedron.The Hilbert series then equals 1 + T ( − T )( − T )( − T ) whence it is easily seen that B ( A , C ) = h , , , i and consequently b ( A ) ≥ D ( B ( A , C )) =
15. Note that this lower bound is stronger than what we could getfrom b ( G ) ≥ max H ( G b ( H ) , since b ( H ) ≤ | H | ≤
12 for any proper subgroup H of A . In Subsection 5.1 we compare known reduction lemmas for the Noether numberwith reduction lemmas for the Davenport constants achieved in previous sections.We demonstrate how to use them to determine the precise value of Noether num-bers and Davenport constants in new examples. In Subsection 5.2 we consider an nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 45 invariant theoretic analogue of the constant h ( G ) (for the definition of h ( G ) see thediscussions before Proposition 2.8 and Lemma 3.1). Throughout this section, suppose that char ( F ) ∤ | G | . In the non-abelian case no structural connection (like Proposition 4.7) is knownbetween the G -invariant polynomials and the product-one sequences over G . Nev-ertheless, a variety of features of the k th Noether numbers and the k th Davenportconstants are strikingly similar, and we offer a detailed comparison.Recall that b k ( G ) = b k ( G ) + d k ( G ) + ≤ D k ( G ) (Proposi-tion 2.8.1).1. The inequalities ( a ) b k ( G ) ≤ k b ( G ) ( b ) d k ( G ) + ≤ k ( d ( G ) + ) ( c ) D k ( G ) ≤ k D ( G ) (7)2. Reduction lemma for normal subgroups N ⊳ G : ( a ) b k ( G ) ≤ b b k ( G / N ) ( N ) ( b ) d k ( G ) ≤ d d k ( N )+ ( G / N ) (8)3. Reduction lemma for arbitrary subgroups H ≤ G with index l = [ G : H ] : ( a ) b k ( G ) ≤ b kl ( H ) ≤ l b k ( H ) ( b ) d k ( G ) + ≤ l ( d k ( H ) + ) ( c ) D k ( G ) ≤ l D k ( H ) (9)4. Supra-additivity: for a normal subgroup N ⊳ G we have ( a ) b k + r − ( G ) ≥ b k ( N ) + b r ( G / N ) if G / N is abelian (10) ( b ) d k + r − ( G ) ≥ d k ( N ) + d r ( G / N )
5. Monotonicity: for an arbitrary subgroup H ≤ G we have ( a ) b k ( G ) ≥ b k ( H ) ( b ) d k ( G ) ≥ d k ( H ) ( c ) D k ( G ) ≥ D k ( H ) (11)6. Almost linearity in k : there are positive constants C , C ′ , C ′′ , k , k ′ , k ′′ dependingonly on G such that ( a ) b k ( G ) = k s ( G ) + C for all k > k if char ( F ) = ( b ) d k ( G ) = k e ( G ) + C ′ (12)for all k > k ′ and ( c ) D k ( G ) = k e ( G ) + C ′′ for all k > k ′′
7. The following functions are non-increasing in k : ( a ) b k ( G ) / k if char ( F ) = ( b ) D k ( G ) / k (13)The inequality (7) (a) is observed in [12], (b) is shown in Proposition 3.9.4,whereas (c) is observed in the beginning of Subsection 2.5.For the proof of (8) (a) see [12, Lemma 1.5] and for part (b) see Proposition 3.9.2.Note that the roles of N and G / N are swapped in the formulas (a) respectively (b),but in the abelian case they amount to the same.The first inequality in part (a) of (9) is proved in [12, Corollary 1.11] for caseswhen (i) char ( F ) = ( F ) > [ G : H ] ; (ii) H is normal in G and char ( F ) ∤ [ G : H ] ; (iii) char ( F ) does not divide | G | . It is conjectured, however that it holds infact whenever char ( F ) ∤ [ G : H ] (see [55]). By [11, Lemma 4.3], we have b kl ( H ) ≤ l b k ( H ) for all positive integers k , l , implying the second inequality in part (a). Parts(b) and (c) of (9) appear in Proposition 3.9 (3. and 5.)Part (a) of (10) appears in [13, Theorem 4.3 and Remark 4.4] while part (b) isproved in Proposition 3.9.1.Parts (b) and (c) of (11) are immediate from the definitions, while part (a) fol-lows from an argument of B. Schmid ([73, Proposition 5.1]) which also shows that b k ( G , Ind GH V ) ≥ b k ( H , V ) for all k ≥ s ( G ) will bediscussed in Subsection 5.2, and for (12) (b) and (c) we refer to Proposition 2.7.2and Proposition 2.8.2.Part (a) of (13) is proved in [11, Section 4] and for (13) (b) we refer to Proposi-tion 2.7.3.Furthermore, for a normal subgroup N ⊳ G we have ( a ) b ( G ) ≤ b ( G / N ) b ( N ) ( b ) D ( G ) ≤ D ( N ) D ( G / N ) , (14)where in (b) we assume that N ∩ G ′ = { } . Here part (a) is originally due to B.Schmid ([73, Lemma 3.1]) and it is an immediate consequence of (7) (a) and (8) (a)while part (b) is proven in [39, Theorem 3.3].The above reduction lemmas on the Noether numbers are key tools in the proofof the following theorem. Theorem 5.1.
Let k ∈ N .1. b k ( A ) = k + and b ( ˜ A ) = , where A is the alternating group of degree and ˜ A is the binary tetrahedral group.2. If G is a non-cyclic group with a cyclic subgroup of index two, then b k ( G ) = | G | k + ( if G = Dic m , m > otherwise.where Dic m = h a , b : a m = , b = a m , bab − = a − i is the dicyclic group.3. nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 47 b ( G ) ≥ | G | if and only if G has a cyclic subgroup of index at most two orG is isomorphic to C ⊕ C , C ⊕ C ⊕ C , A or ˜ A Proof.
For 1. see [12, Theorem 3.4 and Corollary 3.6], for 2. see [13, Theorem 10.3],and 3. can be found in [12, Theorem 1.1].It is worthwhile to compare Theorem 5.1.3 with the statement from [65] assertingthat d ( G ) < | G | unless G has a cyclic subgroup of index at most two. If G is abelian,then Lemma 3.13 and Proposition 4.7 imply d ( G ) + = b ( G ) = D ( G ) . CombiningTheorems 3.10 and 5.1 we obtain that all groups G having a cyclic subgroup of indexat most two satisfy the inequality d ( G ) + ≤ b ( G ) ≤ D ( G ) . Moreover, for thesegroups b ( G ) = d ( G ) +
1, except for the dicyclic groups, where b ( G ) = d ( G ) + H of order27 we have D ( H ) < b ( H ) . Problem 2.
Study the relationship between the invariants d ( G ) , b ( G ) , and D ( G ) .In particular, • Characterize the groups G satisfying d ( G ) + ≤ b ( G ) . • Characterize the groups G satisfying b ( G ) ≤ D ( G ) .In the following examples we demonstrate how the reduction results presented atthe beginning of this section do work. This allows us to determine Noether numbersand Davenport constants of non-abelian groups, for which they were not knownbefore. Example 5.2.
Let p , q be primes such that q | p − G = C p ⋊ C q . A conjecture at-tributed to Pawale ([81]) states that b ( C p ⋊ C q ) = p + q − q = q = q we have onlyupper bounds in [12], proved using known results related to the Olson constant ofthe cyclic group of order p . Theorem 3.11.3 implies that d ( G ) + = p + q − d ( G ) + b ( G ) .2. In view of the great difficulties related to Pawale’s conjecture it is quite remark-able that we can determine the exact value of the Noether number for the non-abeliansemidirect product C pq ⋊ C q . Indeed, this group contains an index p subgroup iso-morphic to C q ⊕ C q , hence b ( C pq ⋊ C q ) ≤ b p ( C q ⊕ C q ) by (9). By Proposition 4.74. we have b p ( C q ⊕ C q ) = D p ( C q ⊕ C q ) , and finally, D p ( C q ⊕ C q ) = pq + q − b ( C pq ⋊ C q ) ≤ pq + q −
1. The reverse inequality alsoholds, since b ( C pq ⋊ C q ) contains a normal subgroup N ∼ = C pq with G / N ∼ = C q , soby (10) and (3) we have b ( C pq ⋊ C q ) ≥ b ( C pq ) + b ( C q ) − = pq + q −
1. So wehave b ( C pq ⋊ C q ) = pq + q − C pq is anormal subgroup and the corresponding factor group is C q , we have by Proposition3.9.1 that d ( C pq ⋊ C q ) ≥ d ( C pq ) + d ( C q ) = p + q −
2. The reverse inequality d ( C pq ⋊ C q ) ≤ p + q − C pq ⋊ C q contains also a normalsubgroup N ∼ = C p such that G / N ∼ = C q ⊕ C q . Consequently, by Lemma 3.1.2.(a) wehave D ( C pq ⋊ C q ) ≥ d ( C pq ⋊ C q ) + = pq + q − . Example 5.3.
The symmetric group S has a normal subgroup N ∼ = C ⊕ C suchthat S / N ∼ = D . We know that b ( D ) = b ( S ) ≤ b b ( D ) ( C ⊕ C ) = D ( C ⊕ C ) = · + = V be the standard 4-dimensional permutation representation of S andsign : S → {± } the sign character. It is not difficult to prove the algebra isomor-phism F [ V ⊗ sign ] S ∼ = F [ V ] S even ⊕ D F [ V ] S odd where D is the Vandermonde determi-nant in 4 variables, F [ V ] S even is the span of the even degree homogeneous componentsof F [ V ] S , and F [ V ] S odd is the span of the odd degree homogeneous components of F [ V ] S . Moreover, the algebra F [ V ] S even ⊕ D F [ V ] S odd is easily seen to be minimallygenerated by s , s , s s , s , s , s D , s D , where s i is the i -th elementary sym-metric polynomial. As a result b ( S , V ⊗ sign ) = deg ( s D ) = + (cid:0) (cid:1) =
9. So weconclude that b ( S ) = Example 5.4.
Let G be the group generated by the complex Pauli matrices (cid:18) (cid:19) , (cid:18) − ii (cid:19) , (cid:18) − (cid:19) . This is a pseudoreflection group, hence the ring of invariants on V = C is gener-ated by two elements, namely C [ x , y ] G = C [ x + y , x y ] . Moreover, b ( G , V ) is thesum of the degrees of the generators minus dim ( V ) (again because G is a pseudore-flection group, see [9]), so b ( G , V ) =
6. It follows by (3) that b ( G ) = b ( G ) + ≥ b ( G , V ) + = G is a non-abelian semi-direct product ( C ⊕ C ) ⋊ C . There-fore G has a normal subgroup N such that N ∼ = G / N ∼ = C ⊕ C and thus b ( G ) ≤ b b ( C ⊕ C ) ( C ⊕ C ) = D ( C ⊕ C ) = . So we conclude that b ( G ) = The constants s ( G , V ) and h ( G , V ) Definition 5.5.
1. Let s ( G , V ) denote the smallest d ∈ N ∪ { ¥ } such that F [ V ] G is a finitely gen-erated module over a subring F [ f , . . . , f r ] such that max { deg ( f i ) : i ∈ [ , r ] } = d .We define s ( G ) = sup { s ( G , W ) : W is a G -module } . nvariant Theory, Multiplicative Ideal Theory, and Arithmetic Combinatorics 49
2. Let S ⊂ F [ V ] G be the F -subalgebra of F [ V ] G generated by its elements of degreeat most s ( G , V ) . Then h ( G , V ) denotes the maximal degree of generators of F [ V ] G + as an S -module.One motivation to study s ( G , V ) and h ( G , V ) is that by a straightforward induc-tion argument ([11, Section 4]) we have b k ( G , V ) ≤ ( k − ) s ( G , V ) + h ( G , V ) . By [11, Proposition 6.2], s ( C p ⋊ C q ) = p (this is also true in characteristic q , see[18, Proposition 4.5]).If F is algebraically closed, then, by Hilbert’s Nullstellensatz, s ( G , V ) is thesmallest d such that there exist homogeneous invariants of degree at most d whosecommon zero locus is the origin. It is shown in Lemma 5.1, 5.4 and 5.6 of [11](some extensions to the modular case and for linear algebraic groups are given in[18]) that • s ( G ) ≤ s ( G / N ) s ( N ) if N ⊳ G ; • s ( H ) ≤ s ( G ) ≤ [ G : H ] s ( H ) if H ≤ G ; • s ( G ) = max { s ( G , V ) : V is an irreducible G -module } . Proposition 5.6.
Let G be abelian.1. s ( G ) = exp ( G ) = e ( G ) .2. h ( G ) = sup { h ( G , W ) : W is a G-module } .Proof. For 1. see [11, Corollary 5.3]. To prove 2., let T ∈ F ( b G ) with | T | = h ( G ) − T has no product-one subsequence U with | U | ∈ [ , e ( G )] . Let V be theregular representation of G , and denote by S the subalgebra of F [ V ] G generated by itselements of degree at most s ( G ) = e ( G ) . Now y : M → F ( b G ) is an isomorphism(see the proof of Proposition 4.7.3.). Thus y − ( T ) ∈ M is not divisible by a G -invariant monomial of degree smaller than e ( G ) . Since both S and F [ V ] are spannedby monomials, it follows that y − ( T ) ∈ M is not contained in the S -submodule of F [ V ] G + generated by elements of degree less than deg ( y − ( T )) . This shows that forthe regular representation V of G we have h ( G , V ) ≥ h ( b G ) .On the other hand let W be an arbitrary G -module, and m ∈ M a monomial withdeg ( m ) > h ( G ) . Then y ( m ) has a product-one subsequence with length at most e ( G ) = s ( G ) , hence m is divisible by a G -invariant monomial of length at most s ( G ) (see the beginning of the proof of Proposition 4.7.2). This shows the inequality h ( G , W ) ≤ h ( b G ) . Taking into account the isomorphism b G ∼ = G we are done.For the state of the art on h ( G ) (in the abelian case) we refer to [40, Theorem5.8.3], [22, 23]. Proposition 5.6 inspires the following problem. Problem 3.
Let G be a finite non-abelian group. Is sup { h ( G , W ) : W is a G -module } finite? Is it related to h ( B ( G )) (see Subsection 2.5 and 3.1)? Acknowledgements
This work was supported by the
Austrian Science Fund FWF (Project No.P26036-N26) and by OTKA K101515 and PD113138.0 K´alm´an Cziszter and M´aty´as Domokos and Alfred Geroldinger
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