Coset closure of a circulant S-ring and schurity problem
aa r X i v : . [ m a t h . C O ] A p r COSET CLOSURE OF A CIRCULANT S-RINGAND SCHURITY PROBLEM
SERGEI EVDOKIMOV AND ILYA PONOMARENKO
Abstract.
Let G be a finite group. There is a natural Galois correspondencebetween the permutation groups containing G as a regular subgroup, and theSchur rings (S-rings) over G . The problem we deal with in the paper, is tocharacterize those S-rings that are closed under this correspondence, when thegroup G is cyclic (the schurity problem for circulant S-rings). It is provedthat up to a natural reduction, the characteristic property of such an S-ring isto be a certain algebraic fusion of its coset closure introduced and studied inthe paper. Basing on this characterization we show that the schurity problemis equivalent to the consistency of a modular linear system associated with acirculant S-ring under consideration. As a byproduct we show that a circulantS-ring is Galois closed if and only if so is its dual. Introduction A Schur ring or S-ring over a finite group G can be defined as a subring of thegroup ring Z G that is a free Z -module spanned by a partition of G closed undertaking inverse and containing { G } as a class (see Section 2 for details). It is wellknown that there is a Galois correspondence between the permutation groups on G that contain the regular group G right , and the S-rings over G :(1) { Γ ≤ Sym( G ) : Γ ≥ G right } ⇄ {A ≤ Z G : A is an S-ring over G } . More precisely, the ” → ” mapping is given by taking the partition of G into theorbits of the stablizer of 1 G in Γ, whereas the ” ← ” mapping is given by taking theautomorphism group of the colored Cayley graph corresponding to the partitionof G associated with A . The Galois closed objects are called 2-closed groups and schurian S-rings, respectively. The schurity problem consists in finding an innercharacterization of schurian S-rings.The theory of S-rings was initiated by I. Schur (1933) and later developed byH. Wielandt [17] and his followers. The starting point for Schur was the Burnsidetheorem stating that any primitive permutation group containing a regular cyclic p -group of composite order, is 2-transitive. Using the S-ring method introduced byhim, Schur generalized this theorem to an arbitrary finite cyclic group G (cf. The-orem 5.1). To some extent this explains the fact that ”Schur had conjecturedfor a long time that every S-ring over G is determined by a suitable permutationgroup” [18, p.54]. This statement had been known as the Schur-Klin conjectureup to 2001, when the first examples of circulant (i.e. over a cyclic group) S-ringswere constructed in [3] by the authors. A recent result in [10] shows that schurian The second author was partially supported by RFFI Grant 14-01-00156. We recall that a Galois correspondence between two posets consists of two mappings reversingthe orders such that both superpositions are closure operators. circulant S-rings are relatively rare. In this paper we provide a solution to theschurity problem for circulant S-rings.The non-schurian examples of S-rings were constructed using the operation of generalized wreath product introduced in [3] (and independently in [13] under thename ”wedge product”). This is not suprising due to the seminal Leung-Mantheorem according to which any circulant S-ring can be constructed from S-ringsof rank 2 and cyclotomic S-rings by means of two operations: tensor product andgeneralized wreath product [13]. Here under a cyclotomic
S-ring A we mean thering of all K -invariant elements of Z G where K is a subgroup of Aut( G ):(2) A = ( Z G ) K . The Leung-Man theorrm reduces the schurity problem for circulant S-rings to find-ing a criterion for the schurity of the generalized wreath product. Such a criterion,based on a generalization of the Leung-Man theory (see [4]), was obtained in pa-per [9] where the generalized wreath product of permutation groups was introducedand studied. All these results form a background to prove the main results of thepaper (see Sections 2–5).Let A be a circulant S-ring. Suppose that among the ”bricks” in the Leung-Mandecomposition of A , there is a non-cyclotomic S-ring. Then this ring is of rank 2, itsunderlying group has composite order and it is Cayley isomorphic to the restrictionof A to one of its sections. Moreover, as it was proved in [9] the S-ring A has aquite a rigid structure that enables us to control the schurity of A . This providesa reduction of the schurity problem to the case when A has no rank 2 section ofcomposite order. The S-rings satisfying the latter property are quasidence in senseof paper [10] (Theorem 5.5). Thus without loss of generality we concentrate on theschurity problem for quasidence S-rings.Our first step is to represent the schurian closure Sch( A ) of a quasidense cir-culant S-ring A in a regular form (Theorem 1.2). The idea here is to replace thering A by a simpler one keeping the structure of its Leung-Man decomposition.The simplification is achieved by changing each ”brick” for a group ring. Thisleads to the class of coset S-rings, i.e. ones for which any class of the correspondingpartition of the group G is a coset of a subgroup in G . It appears that this class isclosed under restriction to a section, tensor and generalized wreath products, andconsists of schurian quasidense S-rings (Theorems 8.4 and 9.1). The regular formof Sch( A )we want to come, will be defined by means of the following concept. Definition 1.1.
The coset closure of a quasidense circulant S-ring A is the inter-section A of all coset S-rings over G that contain A . The coset closure of any quasidense circulant S-ring is a coset S-ring (Theo-rem 8.5). Now, to clarify how to represent the schuran closure of A via its cosetclosure, suppose that the group G is of prime order. In this case it is well knownthat the S-ring A is of the form (2), and, moreover, A = Z G . In particular, A isschurian and any automorphism of G induces a similarity of A . Furthermore, ifthe automorphism belongs to the group K , the similarity is identical on A . Thus A = ( Z G ) K = ( A ) Φ Under a similarity of an S-ring A we mean a ring isomorphism of it that respects the partitionof G corresponding to A , see Subsection 3.1. OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 3 where Φ = Φ ( A ) is the group of all similarities of A that are identical on A . Itappears that this idea works for any quasidense S-ring A . Theorem 1.2.
Let A be a quasidense circulant S-ring. Then Sch( A ) = ( A ) Φ . In particular, A is schurian if and only if A = ( A ) Φ . Theorem 1.2 gives a necessary and sufficient condition for an S-ring to be schurian.This condition being a satisfactory from the theoretical point of view, is hardly aninner characterization. To obtain the latter, we prove Theorem 1.3 below. Let usdiscuss briefly the idea behind it.One of the key properties of coset S-rings that is used in the proof of Theorem 1.2,is that every similarity of any such ring is induced by isomorphism (Theorem 9.1).This fact also shows that in the schurian case the set of all isomorphisms of A that induce similarities belonging to Φ , forms a permutation group the associatedS-ring of which coincides with A . In general, this is not true. A rough reason forthis can be explained as follows. Set(3) S = { S ∈ S ( A ) : ( A ) S = Z S } where S ( A ) is the set of all A -sections and ( A ) S is the restriction of A to S .Then in the schurian case every S-ring A S with S ∈ S , must be cyclotomic,whereas in general this condition does not necessarily hold. However, if even allthe S-rings A S are cyclotomic, one still might find a section S for which A S =Sch( A ) S . These two reasons are controlled respectively by conditions (1) and (2)of Theorem 1.3.It should be mentioned that the proof of the fact that the circulant S-rings con-structed in [3] are non-schurian, was based on studying the relationship betweentheir cyclotomic sections. More careful analysis can be found in [5] where theisomorphism problem for circulant graphs was solved. In that paper the authorsintroduce and study the notion of projective equivalence on the sections of a circu-lant S-ring (this notion is similar to one used in the lattice theory). It appears thatthe class S defined in (3) is closed under the projective equivalence and takingsubsections (Corollary 10.10). Moreover, S ( A ) = S ( A )(statement (1) of Theorem 10.11).To formulate Theorem 1.3 we need additional notation. For S ∈ S ( A ) denoteby Aut A ( S ) the subgroup of Aut( S ) that consists of all Cayley automorphisms ofthe S-ring A S . A family Σ = { σ S } S ∈ S is called a multiplier of A if for any sections S, T ∈ S such that T is projectivelyequivalent to a subsection of S the automorphisms σ T ∈ Aut( T ) and σ S ∈ Aut( S )are induced by raising to the same power . The set of all multipliers of A forms asubgroup of the direct product Q S ∈ S Aut A ( S ) that is denoted by Mult( A ). We recall that any automorphism of a finite cyclic group is induced by raising to a powercoprime to the order of this group.
SERGEI EVDOKIMOV AND ILYA PONOMARENKO
Theorem 1.3.
A quasidense circulant S-ring A is schurian if and only if thefollowing two conditions are satisfied for all S ∈ S : (1) the S-ring A S is cyclotomic, (2) the restriction homomorphism from Mult( A ) to Aut A ( S ) is surjective. By Theorem 10.5 the class S consists of all A -sections S such that each Sylowsubgroup of S (treated as a section of G ) is projectively equivalent to a subsectionof a principal A -section. Thus in contrast to Theorem 1.2, Theorem 1.3 gives anecessary and sufficient condition for an S-ring A to be schurian in terms of A itself rather than of its coset closure A . It should be remarked that in general theclass S may contain non-cyclotomic sections. However, we do not know whethercondition (1) in Theorem 1.3 is implied by condition (2).In fact, the starting point of this paper was the following question: is the propertyof an S-ring ”to be schurian” preserved under taking the dual. The followingtheorem deduced in Section 13 from Theorem 1.3, answers this question in thepositive. Theorem 1.4.
A circulant S-ring is schurian if and only if so is the S-ring dualto it.
We would like to reformulate Theorem 1.3 in the number theoretical language.In what follows we assume that condition (1) of that theorem is satisfied. To makecondition (2) more clear let us fix a section S ∈ S and an integer b coprime to n S = | S | for which the mapping s s b , s ∈ S , belongs to Aut A ( S ). Let usconsider the following system of linear equations in integer variables x S , S ∈ S :(4) ( x S ≡ x T (mod n T ) ,x S ≡ b (mod n S )where S and T run over S and the section T is projectively equivalent to a sub-section of S . We are interested only in the solutions of this system that satisfy theadditional condition(5) ( x S , n S ) = 1 for all S ∈ S . Every such solution produces the family Σ = { σ S } where σ S is the automorphismof the group S taking s to s x S . Moreover, the equations in the first line of (4)guarantee that if a section T is projectively equivalent to a subsection of S , thenthe automorphisms σ T ∈ Aut( T ) and σ S ∈ Aut( S ) are induced by raising to thesame power. Therefore, Σ ∈ Mult( A ) . Conversely, it is easily seen that given S ∈ S every multiplier of A produces asolution of system (4) for the corresponding b . Finally, the consistency of thissystem for all S and all possible b is equivalent to the surjectivity of the restrictionhomomorphism from Mult( A ) to Aut A ( S ) for all S . Thus we come to the followingcorollary of Theorem 1.3. Corollary 1.5.
Let A be a quasidense circulant S-ring such that for any section S ∈ S , the S-ring A S is cyclotomic. Then A is schurian if and only if system (4) has a solution satisfying (5) for all possible S and b . OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 5
Corollary 1.5 reduces the schurity problem for circulant S-rings to solving mod-ular linear system (4) under restriction (5). One possible way to solve this systemis to represent the group Q S Aut A ( S ) as a permutation group on the disjoint unionof the sections S . Then every equation in the first line of (4) defines a subgroup ofthat group the index of which is at most n . Therefore the set of solutions can befound by a standard permutation group technique, see [14, p. 144] for details.Concerning permutation groups we refer to [2]. For the reader convenience anextended background on the S-ring theory including new material, is given in Sec-tions 2–5. In Sections 6 and 7 we study liftings of generalized wreath products inthe non-dense and dense cases respectively. The theory of coset S-rings is developedin Sections 8 and 9, and culminates in Theorem 9.1 showing that these S-rings areschurian and separable. The coset closure and multipliers of a quasidense S-ring areintroduced and studied in Sections 10 and 11 respectively. Theorems 1.2 and 1.3are proved in Section 12. In the final Section 13 we prove Theorem 1.4. Notation.
As usual by Z we denote the ring of rational integers.The set of all right cosets of a subgroup H in a group G is denoted by G/H . Fora set X ⊂ G we put X/H = { Y ∈ G/H : Y ⊂ X } .The subgroup of G generated by a set X ⊂ G is denoted by h X i ; we also setrad( X ) = { g ∈ G : gX = Xg = X } .For a prime p a Sylow p -subgroup of G is denoted by G p .For a section S = U/L of G the quotient epimorphism from U onto S is denotedby π S .For a bijection f : G → G ′ and a set X ⊂ G or an element X ∈ G , the inducedbijection from X onto X f is denoted by f X .The group of all permutations of G is denoted by Sym( G ).For a set ∆ ⊂ Sym( G ) and a section S of G we set∆ S = { f S : f ∈ ∆ , S f = S } . The subgroup of Sym( G ) induced by right multiplications of G is denoted by G right .For X, Y ∈ G/H we set G X → Y = { f X : f ∈ G right , X f = Y } .The holomorph Hol( G ) is identified with the subgroup of Sym( G ) generated by G right and Aut( G ).The orbit set of a group Γ ≤ Sym( G ) is denoted by Orb(Γ) = Orb(Γ , G ).We write Γ ≈ Γ ′ if groups Γ , Γ ′ ≤ Sym( G ) are 2-equivalent, i.e. have the sameorbits in the coordinate-wise action on G × G .For a positive integer n , the cyclic group the elements of which are integersmodulo n , is denoted by Z n .For an automorphism σ of a cyclic group G of order n the unique element m ∈ Z n for which x σ = x m , x ∈ G , is denoted by m ( σ ).2. S-rings
In what follows we use the notation and terminology from paper [9] where a partof the material is contained. Concerning the basic S-ring theory and duality theorywe also refer to [1, Ch.2.6] and [17, Ch.4], respectively.
SERGEI EVDOKIMOV AND ILYA PONOMARENKO
Definitions and basic facts.
Let G be a finite group with identity 1 G . Asubring A of the group ring Z G is called a Schur ring ( S-ring , for short) over G ifthere exists a partition S ( A ) of G such that(1) { G } ∈ S ( A ),(2) X ∈ S ( A ) = ⇒ X − ∈ S ( A ),(3) A = span { P x ∈ X x : X ∈ S ( A ) } .Two S-rings A and A ′ are called Cayley isomorphic if there exists a ring isomor-phism from A onto A ′ that is induced by isomorphism of underlying groups; thelatter is called Cayley isomorphism from A onto A ′ . Obviously, S ( A ) f = S ( A ′ ) forany such isomorphism f .The elements of S ( A ) and the number rk( A ) = |S ( A ) | are called respectivelythe basic sets and the rank of the S-ring A . Any union of basic sets is called an A -subset of G or A -set . The set of all of them is closed with respect to takinginverse and product, and forms a lattice with respect to inclusion. Given an A -set X the submodule of A spanned by the set S ( A ) X = { Y ∈ S ( A ) : Y ⊂ X } is denoted by A X .A subgroup of G that is an A -set is called A -subgroup of G or A -group ; the setof all of them is denoted by G ( A ). With each A -set X one can naturally associatetwo A -groups, namely h X i and rad( X ) (see Notation). The S-ring A is called dense if every subgroup of G is an A -group, and primitive if the only A -subgroups are 1and G .A section S = U/L of the group G is called a section of A or an A -section ,if both U and L are A -groups. In this case we also say that A contains S . Themodule A S = span { π S ( X ) : X ∈ S ( A ) U } is an S-ring over the group S , the basic sets of which are exactly the sets in theright-hand side of the formula. A section S is of rank 2 (resp. primitive, cyclotomic)if so is the S-ring A S . The set of all (resp. all cyclotomic) A -sections is denoted by S ( A ) (resp. S cyc ( A )). Set S prin ( A ) = {h X i / rad( X ) : X ∈ S ( A ) } . Any element of this set is called a principal A -section.The partial order on the set of all S-rings over G that is induced by inclusion, isdenoted by “ ≤ ”. Thus A ≤ A if and only if any basic set of A is a union ofbasic sets of A (in this case we say that A is an extension of A ). The least andgreatest elements are span { , X x ∈ G x } and Z G respectively. Next, the module A ∩ A is an S-ring, the basic sets of which formthe finest partition of G that is coarser than both S ( A ) and S ( A ). It follows thatthe set of all S-rings over G forms a lattice in which the meet (resp. join) of A and A coincides with A ∩ A (resp. with the intersection of all S-rings over G that contain both A and A ). It is easily seen that(6) ( A ∩ A ) S = ( A ) S ∩ ( A ) S OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 7 for any section S ∈ S ( A ) ∩ S ( A ).Let K ≤ Aut( G ) be a group of Cayley isomorphisms of the S-ring A . Thenthe set A K of the elements in A left fixed under the induced action of K on Z G ,forms an S-ring over G . Any such ring with A = Z G , is called cyclotomic and isdenoted by Cyc( K, G ). The classes of the corresponding partition of G are exactlythe orbits of the group K .If A and A are S-rings over groups G and G respectively, then the subring A = A ⊗ A of the ring Z G ⊗ Z G = Z G where G = G × G , is an S-ring overthe group G with S ( A ) = { X × X : X ∈ S ( A ) , X ∈ S ( A ) } . It is called the tensor product of A and A .2.2. Generalized wreath product.
Let S = U/L be a section of an S-ring A .We say that A is an S -wreath product if the group L is normal in G and L ≤ rad( X )for all basic sets X outside U ; in this case we write(7) A = A U ≀ S A G/L , and omit S when | S | = 1; in the latter case A is called wreath product . When anexplicit reference to the section S is not important, we use the term generalizedwreath product . The S -wreath product is nontrivial or proper if 1 = L and U = G . Theorem 2.1.
Let S = U/L be a section of a group G , and let A and A beS-rings over the groups U and G/L respectively such that S is both an A - and an A -section with ( A ) S = ( A ) S . Then the set of S-rings A such that A U = A and A G/L = A has the smallestelement. Moreover, it is a unique S -wreath product in this set. Proof . It was proved in [3, Theorem 3.1] that the S -wreath product A from (7) isuniquely determined and belongs to the set of S-rings from the theorem statement.For any other S-ring A ′ from this set the preimage in S ( A ′ ) of at least one basic setin S ( A ) outside U/L contains at least two basic sets, whereas the same preimage in S ( A ) consists of one element. This proves that A ′ ≥ A , and hence the minimalityof A .2.3. Duality.
Let A be an S-ring over a finite abelian group G and b G the groupdual to G , i.e. the group of all irreducible complex characters of G . Given S ⊂ G and χ ∈ b G set(8) χ ( S ) = X s ∈ S χ ( s ) . Characters χ , χ ∈ b G are called equivalent if χ ( S ) = χ ( S ) for all S ∈ S ( A ).Denote by b S the set of classes of this equivalence relation. Then the submodule of Z b G spanned by the elements ξ ( X ), X ∈ b S , is an S-ring over b G (see [1, Theorem 6.3]).This ring is called dual to A and is denoted by b A . Obviously, S ( b A ) = b S . Moreover,rk( b A ) = rk( A ) and(9) G ( b A ) = { H ⊥ : H ∈ G ( A ) } SERGEI EVDOKIMOV AND ILYA PONOMARENKO where H ⊥ = { χ ∈ b G : H ≤ ker( χ ) } . It is also true that the S-ring dual to b A isequal to A .It is easily seen that the mapping from Aut( G ) to Aut( b G ) that takes σ to b σ defined by χ b σ ( g ) = χ ( g σ ), is a group isomorphism. The image of a group K ≤ Aut( G ) is denoted by b K . Theorem 2.2.
Let A = Cyc( K, G ) where K ≤ Aut( G ) . Then b A = Cyc( b K, b G ) . Proof . Let X ∈ Orb( b K, b G ). Then given χ , χ ∈ X there exists σ ∈ K suchthat χ = χ b σ . Since S = S σ for each basic set S of A , this implies that χ ( S ) = χ b σ ( S ) = χ ( S σ ) = χ ( S ) , S ∈ S ( A ) . Therefore Cyc( b K, b G ) ≥ b A . Replacing here K and G by b K and b G respectively, weconclude that the S-ring A contains the S-ring dual to Cyc( b K, b G ). By duality thisimplies that b A ≥
Cyc( b K, b G ). Thus b A = Cyc( b K, b G ).Some more properties of the dual S-ring are contained in the following statementproved in [11, Theorems 2.4, 2.5]. In what follows given a section S = U/L thegroup b S is canonically identified with the section L ⊥ /U ⊥ of the group b G that iscalled the section dual to S . In particular, if G = G × G , then b G = c G × c G . Theorem 2.3.
Let A be an S-ring over an abelian group G . Then (1) c A S = b A b S for any S ∈ S ( A ) , (2) A = A ⊗ A if and only if b A = b A ⊗ b A , (3) A is an S -wreath product if and only if b A is an b S -wreath product. Similarities, isomorphisms and schurity
In this section we follow papers [3, 4, 9] except for terminology: similarities andisomorphisms defined below were called in [3, 4] weak and strong isomorphismsrespectively (see also the remark in Subsection 3.2). The reader familiar withassociation scheme theory will see that the definitions of similarity, isomorphism,etc. given in this section, are compatible with those used for Cayley schemes(see [9]).3.1.
Similarities.
Let A and A ′ be S-rings over groups G and G ′ respectively. Aring isomorphism ϕ : A → A ′ is called similarity from A to A ′ , if for any X ∈ S ( A )there exists X ′ ∈ S ( A ′ ) such that ϕ ( X x ∈ X x ) = X x ′ ∈ X ′ x ′ . It follows from the definition that the mapping X X ′ is a bijection from S ( A )onto S ( A ′ ). This bijection is naturally extended to a bijection between A - and A ′ -sets, that takes G ( A ) to G ( A ′ ), and hence S ( A ) to S ( A ′ ). The images of an A -set X and A -section S are denoted by X ϕ and S ϕ respectively. For any such S the similarity ϕ induces a similarity ϕ S : A S → A ′ S ′ where S ′ = S ϕ . The set of all similarities from A to A ′ is denoted by Φ( A , A ′ ); we also set Φ( A ) =Φ( A , A ). OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 9
The above bijection between the A - and A ′ -sets is in fact an isomorphism of thecorresponding lattices. It follows that given an A -set X we have(10) h X ϕ i = h X i ϕ and rad( X ϕ ) = rad( X ) ϕ . These equalities together with the obvious equation X = X rad( X ), immediatelyimply the following statement. Lemma 3.1.
Any similarity of an S-ring is uniquely determined by its restrictionsto principal sections.
Any automorphism σ ∈ Aut( G ) can be extended linearly to a ring automor-phism ϕ σ of Z G . Thus the lemma below immediately follows from the definition ofsimilarity. Lemma 3.2.
The mapping σ ϕ σ is a group isomorphism from Aut( G ) onto Φ( Z G ) . Let Φ be a group of similarities of the S-ring A . Then the set A Φ of the elementsin A left fixed under the action of Φ, is obviously an S-ring over G for which S ( A Φ ) = { X Φ : X ∈ S ( A ) } where X Φ = S ϕ ∈ Φ X ϕ . When A = Z G , from Lemma 3.2 it follows that A Φ is acyclotomic S-ring.We complete the subsection by describing the similarities of generalized wreathproducts; mostly this was done in [3]. Theorem 3.3.
Let A and A ′ be S-rings over abelian groups G and G ′ . Supposethat A is an S -wreath product where S = U/L . Then (1) for any similarity ϕ ∈ Φ( A , A ′ ) the S-ring A ′ is the S ′ -wreath product where S ′ = S ϕ , (2) if A ′ is an S ′ -wreath product, then the mapping ϕ ( ϕ U , ϕ G/L ) induces abijection from the set { ϕ ∈ Φ( A , A ′ ) : S ′ = S ϕ } to the set { ( ϕ , ϕ ) ∈ Φ( A U , A ′ U ′ ) × Φ( A G/L , A ′ G ′ /L ′ ) : ( ϕ ) S = ( ϕ ) S } . where U ′ = U ϕ and L ′ = L ϕ . Proof . Statement (2) follows from [3, Theorem 3.3] . To prove statement (1)it suffices to verify that L ′ X ′ = X ′ for all X ′ ∈ S ( A ′ ) G ′ \ U ′ . However, this is truebecause ( G \ U ) ϕ = G ′ \ U ′ and LX = X for all X ∈ S ( A ) G \ U .3.2. Isomorphisms.
Let A and A ′ be S-rings over groups G and G ′ respectively.A bijection f : G → G ′ is called an isomorphism from A onto A ′ if there exists asimilarity ϕ ∈ Φ( A , A ′ ) such that given X ∈ S ( A ) we have(11) f ( Xy ) = X ϕ f ( y ) for all y ∈ G, or, equivalently, f ( x ) f ( y ) − ∈ X ϕ for all x, y ∈ G with xy − ∈ X . In this case wealso say that f induces ϕ . Clearly, any isomorphism induces a uniquely determinedsimilarity. The set of all isomorphisms and isomorphisms with a fixed ϕ are denotedby Iso( A , A ′ ) and Iso( A , A ′ , ϕ ) respectively.It follows from the definition that the isomorphism f that takes 1 G to 1 G ′ , takes S ( A ) to S ( A ′ ) and satisfies the condition f ( Xy ) = f ( X ) f ( y ) for all X ∈ S ( A )and y ∈ G . Therefore f is a strong isomorphism from A to A ′ in the sense of [3]. Conversely, according to that paper any strong isomorphism f induces a similarity ϕ such that f ( X ) = X ϕ , and hence satisfies (11). Thus S-rings are isomorphic ifand only if they are strongly isomorphic (see equality (12) below).It immediately follows from the definition that(12) G right Iso( A , A ′ , ϕ ) G ′ right = Iso( A , A ′ , ϕ ) . In particular, not every isomorphism takes 1 G to 1 G ′ . But even if it does, it is notnecessarily a Cayley isomorphism. However, | G | = | G ′ | because every similaritypreserves the order of the underlying group. Moreover, since equality (11) obviouslyholds also for any A -set X , and, in particular, every isomorphism preserves the rightcosets of any A -group. Lemma 3.4.
In the above notation let H be an A -group and H ′ = H ϕ . Then hf X h ′ ∈ Iso( A H , A ′ H ′ , ϕ H ) for all X ∈ G/H , h ∈ G H → X and h ′ ∈ G ′ X ′ → H ′ where X ′ = X f . Proof . Denote by g and g ′ the permutations from G right and G ′ right such that g H = h and ( g ′ ) X ′ = h ′ , respectively. Then by (12) the bijection gf g ′ : G → G ′ induces the similarity ϕ . So the bijection hf X h ′ = ( gf g ′ ) H induces the similarity ϕ H as required.The following statement characterizes the isomorphisms of a generalized wreathproduct. Theorem 3.5.
Let A and A ′ be S-rings over abelian groups G and G ′ and ϕ : A →A ′ a similarity. Suppose that A is a U/L -wreath product. Then the set
Iso( A , A ′ , ϕ ) consists of all bijections f : G → G ′ such that ( G/U ) f = G ′ /U ′ , ( G/L ) f = G ′ /L ′ where U ′ = U ϕ and L ′ = L ϕ , and (13) f G/L ∈ Iso( A G/L , A ′ G/L , ϕ
G/L ) , gf X g ′ ∈ Iso( A U , A ′ U , ϕ U ) for all X ∈ G/U and some g ∈ G U → X and g ′ ∈ G ′ X ′ → U ′ where X ′ = X f . Proof . Set F = Iso( A , A ′ , ϕ ) and denote by F ′ the set of all bijections f : G → G ′ satisfying (13). Then the inclusion F ⊃ F ′ immediately follows from the basicproperties of similarities and Lemma 3.4. Conversely, let f ∈ F ′ . We have to verifythat equality (11) holds for all X ∈ S ( A ) and y ∈ G . Suppose first that X ⊂ G \ U .Then from the equality ( G/L ) f = G ′ /L ′ and the first relation in (13) it follows that f G/L ( X π y π ) = ( X π ) ϕ G/L f G/L ( y π )where π = π G/L and y ∈ G . After taking the preimages of both sides in the latterequality we obtain that f ( XyL ) = ( XL ) ϕ f ( yL ). On the other hand, L ≤ rad( X ).Due to (10) this implies that L ′ ≤ rad( X ϕ ). Thus f ( Xy ) = f ( XyL ) = ( XL ) ϕ f ( yL ) = X ϕ f ( y ) L ′ = X ϕ f ( y ) , which proves the required statement in our case. Let now X ⊂ U and y ∈ G . Thenfrom the equality ( G/U ) f = G ′ /U ′ and the second relation in (13) it follows that X gf Y g ′ = X ϕ for some g ∈ G U → Y and g ′ ∈ G ′ Y ′ → U ′ where Y = U y and Y ′ = Y f . By Lemma 3.4without loss of generality we can assume that the permutations g and g ′ are induced OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 11 by multiplications by y and ( y f ) − respectively. Then X gf Y g ′ = ( Xy ) f ( y f ) − , andhence f ( Xy ) = ( Xy ) f = X ϕ y f = X ϕ f ( y )and we are done.Let K be a class of S-rings closed under Cayley isomorphisms. Following [7] anS-ring A is called separable with respect to K if Iso( A , A ′ , ϕ ) = ∅ for all similarities ϕ : A → A ′ where A ′ ∈ K . In this paper we say that a circulant S-ring is separable if it is separable with respect to the class of all circulant S-rings.3.3. Schurity.
Let G be a finite group. It was proved by Schur (see [17, Theo-rem 24.1]) that any group Γ ≤ Sym( G ) that contains G right produces an S-ring A over G such that S ( A ) = Orb(Γ , G )where Γ = { γ ∈ Γ : 1 γ = 1 } is the stabilizer of the point 1 = 1 G in Γ. Any suchS-ring is called schurian . Group rings and S-rings of rank 2 are obviously schurian.Let Γ and ∆ be permutation groups on G such that G right ≤ Γ ∩ ∆. Then it iseasily seen that the S-ring associated with the group h Γ , ∆ i equals the intersectionof S-rings associated with Γ and ∆. It follows that the intersection of schurianS-rings is schurian. Therefore so is the S-ring(14) Sch( A ) = \ A ′ ≥A , A ′ is schurian A ′ . It is called the schurian closure of A . Clearly, Sch( A ) ≥ A , and the equality isattained if and only if the S-ring A is schurian.The schurity concept is closely related to automorphisms of an S-ring. In con-trast to a common algebraic tradition the automorphism group of an S-ring A isnot defined to be Iso( A , A ); in accordance with a combinatorial tradition we setAut( A ) = Iso( A , A , id A ). Thus f ∈ Aut( A ) if and only if(15) f ( Xy ) = Xf ( y ) , X ∈ S ( A ) , y ∈ G. The latter is equivalent to say that given X ∈ S ( A ) we have f ( x ) f ( y ) − ∈ X when-ever xy − ∈ X . Therefore Aut( A ) can also be defined as the automorphism groupof the colored Cayley graph corresponding to the partition S ( A ) of the group G (cf. Introduction).It follows that any basic set of A is invariant with respect to the group Aut( A ) ,whereas equality (12) shows that G right ≤ Aut( A ). Moreover, the group Aut( A )is the largest subgroup of Sym( G ) that satisfies these two properties. Now, let Γbe a permutation group the schurian S-ring Sch( A ) is associated with. Then sinceSch( A ) ≥ A , the maximality of Aut( A ) implies thatΓ ≤ Aut( A ) . It follows that Sch( A ) contains the S-ring associated with Aut( A ). Therefore theseS-rings are equal (see (14)). In fact, this shows that the closure of the S-ring A with respect to Galois correspondence (1) coincides with Sch( A ). Thus the abovedefinition of a schurian S-ring is compatible with that given in Introduction. Let f ∈ Aut( A ) . Then any A -set (in particular, A -group) is invariant withrespect to the automorphism f . Moreover, for any A -section S we have f S ∈ Aut( A S ). In particular, the S-ring A S is schurian whenever so is A .The following result proved in [9, Corollary 5.7] gives a criterion for the schurityof generalized wreath products that will be repeatedly used throughout the paper.Below we set M ( A ) = { Γ ≤ Sym( G ) : Γ ≈ Aut( A ) and G right ≤ Γ } . Theorem 3.6.
Let A be an S-ring over an abelian group G . Suppose that A isan S -wreath product where S = U/L . Then A is schurian if and only if so are theS-rings A G/L and A U and there exist groups ∆ ∈ M ( A G/L ) and ∆ ∈ M ( A U ) such that (∆ ) S = (∆ ) S . It should be remarked that a permutation f ∈ Sym( G ) preserving every basic setof A does not necessarily belong to Aut( A ). However, as the lemma below shows,this is so when, for example, f ∈ Aut( G ). Lemma 3.7.
Let A be an S-ring and A ′ ≥ A . Then the group of all isomorphismsof A ′ that fix every element of A , is a subgroup of Aut( A ) . Proof . Let f be an isomorphism of A ′ and ϕ the similarity of A ′ induced by f .Then f ( X ′ y ) = ( X ′ ) ϕ f ( y ) for all A ′ -sets X ′ and y ∈ G . Since A ′ ≥ A , this is truefor all X ′ ∈ S ( A ). So if f fixes every element of A , then ϕ is identical on A , andhence f ( Xy ) = Xf ( y ) for all X ∈ S ( A ). Thus f ∈ Aut( A ).4. Sections in S-rings
Projective equivalence.
Let G be a group. Denote by G ( G ) the set ofits subgroups, and by S ( G ) the set of its sections, i.e. quotients of subgroupsof G . When this does not lead to misunderstanding, we write H instead of H/ H ∈ G ( G ), and identify S ( S ) with the corresponding subset of S ( G ) where S ∈ S ( G ).A section U ′ /L ′ is called a subsection of a section U/L if U ′ ≤ U and L ′ ≥ L ;in this case we write U ′ /L ′ (cid:22) U/L . This defines a partial order on the set S ( G ).This order has the greatest element G/
1; any minimal element is of the form
H/H where H ∈ G ( G ).A section U/L is called a multiple of a section U ′ /L ′ if(16) L ′ = U ′ ∩ L and U = U ′ L. The projective equivalence relation ” ∼ ” on the set S ( G ) is defined to be the tran-sitive closure of the relation ”to be a multiple”. Any two projecively equivalentsections are obviously isomorphic as groups. The set of all equivalence classes isdenoted by P ( G ). Under a quasisubsection of a section S we mean any sectionwhich is projectively equivalent to a subsection of S .Given a section S = H/K of a group G we define a surjection ρ G,S from thesubgroups of G to the subgroups of S by ρ G,S ( U ) = ( U ∩ H ) K/K.
OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 13
Let us extend this mapping to S ( G ) by ρ G,S ( U/L ) = ρ G,S ( U ) /ρ G,S ( L ). Using theabove identification of S ( S ) with the corresponding subset of S ( G ), we obtain that(17) ρ G,S ( U/L ) = ( U ∩ H ) K/ ( L ∩ H ) K. Clearly, this mapping is identical on the set S ( S ). Moreover, ρ G,S inducess asurjective homomorphism of the corresponding partially ordered sets. To simplifynotations we will write T S instead of ρ G,S ( T ) when the group G is fixed.If the group G is abelian, then the set G ( G ) of all subgroups of G forms a modularlattice in which the join and meet of H and K are defined as HK and H ∩ K respectively [16]. Lemma 4.1.
Let G be an abelian group. Then given S, T ∈ S ( G ) we have S T ∼ T S . Proof . Let S = H/K and T = U/L . Then by the definition of S T and T S wehave S T = ( H ∩ U ) L/ ( K ∩ U ) L and T S = ( U ∩ H ) K/ ( L ∩ H ) K. A straightforward check shows that both S T and T S are multiples of the section( H ∩ U ) / ( H ∩ U ∩ KL ) (we made use the fact the lattice G ( G ) is modular). Thereforethe sections S T and T S are projectively equivalent.4.2. Restrictions.
Let A be an S-ring over a group G . Set P ( A ) = { C ∩ S ( A ) : C ∈ P ( G ) , C ∩ S ( A ) = ∅} . Then P ( A ) forms a partition of the set S ( A ) into classes of projectively equivalent A -sections. It should be noted that if A ′ ≥ A , then S ( A ′ ) ⊃ S ( A ) and each classof projectively equivalent A -sections is contained in a unique class of projectivelyequivalent A ′ -sections. Theorem 4.2.
Let A be an S-ring over a group G . Then given projectively equiv-alent A -sections S and T there exists a Cayley isomorphism f from A S onto A T such that ( γ S ) f = γ T for all γ ∈ Aut( A ) leaving the point G fixed. Proof . Follows from [9, Lemma 3.1,Theorem 3.2].Obviously, any two sections of a class C ∈ P ( G ) have the same order; we call itthe order of this class. If, in addition, C ∈ P ( A ) where A is an S-ring over G , thenfrom Theorem 4.2 it follows that all sections in C have the same rank r , and alsoif C contains a primitive (resp. cyclotomic, dense) section, then all sections in C are primitive (resp. cyclotomic, dense). In these cases we say that C is a class ofrank r , and a primitive (resp. cyclotomic, dense) class.In general, an A -section projectively equivalent to a principal A -section is notprincipal. However, at least in the cyclic group case such a section is subprincipal ,i.e. a subsection of a principal A -section (Lemma 5.3). But even in this case theclass of subprincipal sections is not closed under the projective equivalence. Toavoid this inconvenience we define an A -section to be quasiprinciple (resp. qua-sisubprinciple ) if it is projectively equivalent to a principal (resp. subprincipal) A -section.Let S ∈ S ( A ). Then the mapping ρ G,S : T T S defined in Subsection 4.1induces a mapping from S ( A ) to S ( A S ) that is denoted by the same letter. Thefollowing statement shows that it preserves generalized wreath products. We recall that a lattice is modular, if x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ z whenever x ≤ z . Theorem 4.3.
Let A be an S-ring over an abelian group G . Suppose that A isthe T -wreath product where T ∈ S ( A ) . Then A S is the T S -wreath product forall S ∈ S ( A ) . Proof . Let T = U/L and S = H/K . We have to prove that L S ≤ rad( Y ) for all Y ∈ A S \ U S . However, any Y ∈ S ( A S ) is of the form XK/K for some X ∈ S ( A H ).If, in addition, Y ∈ A S \ U S , then X ∈ A G \ U , and hence L ≤ rad( X ) because A isthe T -wreath product. Since rad( X ) ≤ h X i ≤ H , this implies that L S ≤ rad( X ) S = rad( X ) K/K ≤ rad( XK/K ) = rad( Y )as required.4.3. Duality.
Let G be an abelian group. Then the mapping H H ⊥ induces alattice antiisomorphism from G ( G ) onto G ( b G ) (see e.g. [16]). Lemma 4.4.
For any sections
S, T ∈ S ( G ) the following statements hold: (1) S ∼ T if and only if b S ∼ b T ; moreover, T is a multiple of S if and only if b S is a multiple of b T , (2) \ ρ G,S ( T ) = ρ b G, b S ( b T ) , (3) S (cid:22) T if and only if b S (cid:22) b T . Proof . Let S = H/K and T = U/L . To prove statement (1) without loss ofgenerality we can assume that T is a multiple of S . Then from (16) it follows that K ⊥ = H ⊥ L ⊥ and U ⊥ = H ⊥ ∩ L ⊥ , which means that b S = K ⊥ /H ⊥ is a multiple of b T = L ⊥ /U ⊥ as required. State-ment (2) follows from (17) and the modularity of the lattice G ( G ): \ ρ G,S ( T ) = (( U ∩ H ) K ) ⊥ / (( L ∩ H ) K ) ⊥ = (( U K ) ∩ H ) ⊥ / (( LK ) ∩ H ) ⊥ =( U ⊥ ∩ K ⊥ ) H ⊥ / ( L ⊥ ∩ K ⊥ ) H ⊥ = ρ b G, b S ( b T ) . Statement (3) is obvious.Let A be an S-ring over G . Given a class C ∈ P ( A ) we define the dual class by b C = { b S : S ∈ C } . Then from statement (1) of Lemma 4.4 it follows that b C ∈ P ( b A ). Moreover, theclasses b C and C have the same order and rank, and if one of them is primitive (resp.cyclotomic, dense), then so is the other one (see equality (9) and Theorem 2.2).5. Circulant S-rings
General theory.
We begin with a well-known result on circulant primitiveS-rings, that goes back to Burnside and Schur. Despite the fact that they dealtwith groups, their results can be interpreted as results on schurian circulant S-rings.Moreover, the Schur method works in the non-schurian case as well (see, e.g. [6]).
Theorem 5.1.
Any circulant primitive S-ring is of rank , or a cyclotomic S-ringover a group of prime order. OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 15
Let A be an S-ring over a cyclic group G . Then X σ ∈ S ( A ) for all X ∈ S ( A ) , σ ∈ Aut( G ) , see [17, Theorem 23.9]. Let now X be a highest basic set of A , i.e. one containinga generator of G . Then the above statement implies that the group rad( X ) doesnot depend on the choice of X . It is called the radical of A and denoted by rad( A ).The following statement proved by Leung and Man is a cornerstone of the circulantS-ring theory (see [4, Corollaries 5.5,6.4]). Theorem 5.2.
Let A be a circulant S-ring. Then the following two statementshold: (1) rad( A ) = 1 if and only if A is a proper generalized wreath product, (2) rad( A ) = 1 if and only if A is the tensor product of a cyclotomic S-ringwith trivial radical and S-rings of rank . Any principal A -section S is obviously a section with trivial radical in the sensethat rad( A S ) = 1. The converse is not true, but we have the following statement. Lemma 5.3.
In a circulant S-ring, any section with trivial radical is subprincipal.
Proof . Suppose that rad( A S ) = 1 where S = U/L is a section of a circulantS-ring A . We observe that every highest basic set X of the S-ring A U produces ahighest basic set π ( X ) of the S-ring A S where π = π S . Thus π (rad( X )) ≤ rad( π ( X )) = rad( A S ) = 1 . It follows that rad( X ) ≤ L . Therefore S (cid:22) T where T = U/ rad( X ). Since U = h X i , the A -section T is principal, and we are done.From Theorem 4.2 it follows that the set of all A -sections with trivial radical isclosed with respect to the projective equivalence. Another property of the projectiveequivalence for circulant S-rings is given in the following statement proved in [5,Lemma 5.2]. Below a section of a class C ∈ P ( A ) is called the smallest (respectively, largest ) one if every section of C , is a multiple of it (respectively, if it is a multipleof every section of C ). Theorem 5.4.
Any class of projectively equivalent sections of a circulant S-ringhas the largest and smallest elements.
Quasidence S-rings.
A circulant S-ring A is called quasidense , if any prim-itive A -section is of prime order. Any dense S-ring is obviously quasidense. More-over, in the quasidense case any minimal A -group is of prime order, any maximal A -group is of prime index, and the S-ring A S is dense for any A -section S of primepower order. Theorem 5.5.
A circulant S-ring is quasidense if and only if it contains no rank sections of composite order. Proof . By Theorem 5.1 a section of composite order is of rank 2 if and only if itis primitive. Thus the required statement immediately follows from the definitionof a quasidence S-ring.It is easily seen that the class of quasidense S-rings is closed under restriction toa section, and the tensor and generalized wreath products.
Theorem 5.6.
Any extsension of quasidense circulant S-ring is quasidense.
Proof . Let A ′ be a non-quasidense extension of a quasidense S-ring A over acyclic group G . Then by Theorem 5.5 the S-ring A ′ contains a rank 2 section S = U/L of composite order. Denote by H an A -group of prime order (such agroup does exist because A is quasidense). We claim that(18) H ∩ U = 1 or H ≤ L. Indeed, if (18) is not true, then H ≤ U and H ∩ L = 1 because the order of H isprime. It follows that the S-ring A ′ S contains the group HL/L of prime order | H | .The latter group does not equal S because the order of S is composite. Howeverthis is impossible because rk( A ′ S ) = 2.By (18) the S-ring A ′ G/H contains the section S ′ = U H/LH which is projectivelyequivalent to the section
U/L by the modularity of the lattice G ( G ). By Theorem 4.2this implies that rk( A ′ S ′ ) = rk( A ′ S ) = 2 . Thus, A ′ G/H is a non-quasidense extension of the quasidense S-ring A G/H due toTheorem 5.5. Assuming without loss of generality that the order of the group G isminimal possible we come to a contradiction. Theorem 5.7.
The class of quasidense circulant S-rings with trivial radical isclosed with respect to taking extensions, and consists of cyclotomic, and hence denseS-rings.
Proof . Let A be a quasidense circulant S-ring with trivial radical. Then it iscyclotomic, and hence dense by [10, Theorem 3.1]. If now A ′ ≥ A , then the S-ring A ′ is dense, and hence quasidense. Suppose on the contrary that rad( A ′ ) = 1. Thenby statement (1) of Theorem 5.2 the S-ring A ′ is a proper S ′ -wreath product forsome A ′ -section S ′ . This section is also A -section by the density of A . Therefore A is the S ′ -wreath product, which is impossible by statement (1) of Theorem 5.2because rad( A ) = 1.The following theorem deduced from [10, Theorem 3.5] shows that any schurianquasidense circulant S-ring can be obtained from an appropriate solvable permuta-tion group that ”locally” has a rather simple form. Theorem 5.8.
Let A be a schurian quasidense circulant S-ring. Then there existsa group Γ ∈ M ( A ) such that Γ S = Hol A ( S ) for any quasisubprinciple A -section S where Hol A ( S ) = Hol( S ) ∩ Aut( A S ) . Proof . By [10, Theorem 3.5] there exists a group Γ ∈ M ( A ) such that Γ S =Hol A ( S ) for any A -section S such that rad( A S ) = 1. Since obviously Hol A ( S ) T =Hol A ( T ) for all A -sections T (cid:22) S , the required statement follows from Theorem 4.2for γ ∈ Γ.5.3.
Duality.
The following two theorems establishing selfdual properties of a cir-culant S-ring will be used in Section 13 to prove Theorem 1.4.
Theorem 5.9.
Let A be a circulant S-ring. Then (1) rad( A ) = 1 if and only if rad( b A ) = 1 , (2) A is quasidense if and only if b A is quasidence. OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 17
Proof . Statement (1) immediately follows from statement (3) of Theorem 2.3and Theorem 5.2. Statement (2) holds because the primitivity and the order of an A -section are preserved under duality: the former by equality (9) and statement (1)of Theorem 2.3, whereas the latter by the definition of the dual section. Theorem 5.10.
A section of a circulant S-ring is quasisubprincipal if and only ifso is its dual.
Proof . Let S be a quasisubprincipal section of a circulant S-ring A . Then S isprojectively equivalent to a subsection S ′ of a principal A -section T . Without lossof generality we can assume that T is a (cid:22) -maximal principal A -section. Then b T is a (cid:22) -maximal b A -section by statement (1) of Lemma 4.4. Moreover, by statement (2)of Theorem 5.9 we have rad( b A b T ) = 1. Thus by Lemma 5.3 we conclude that b T is aprincipal b A -section. On the other hand, from statements (1) and (3) of Lemma 4.4we have b S ∼ b S ′ and b S ′ (cid:22) b T . Thus b S is a quasisubprincipal A -section as required.The converse statement holds by duality.6. Lifting: nondense case
We begin with a characterization of the dense circulant S-rings in terms of for-bidden subsections. To do this we will say that an S-ring A over a cyclic group G is elementary nondense if | G | is a composite number and A has rank 2, or | G | is theproduct of two distinct primes and A is a proper wreath product. In the formercase the S-ring is not quasidense, whereas in the latter case it is. A section S ofa circulant S-ring A is called elementary nondense if the S-ring A S is elementarynondense. Lemma 6.1.
The section that is dual or projectively equivalent to an elementarynondense section is elementary nondense.
Proof . Follows from Theorems 2.3 and 4.2.It is easily seen that a circulant S-ring that contains an elementary nondensesection cannot be dense. This proves the “if” part of the following statement.
Theorem 6.2.
A circulant S-ring A is not dense if and only if there exists anelementary nondense A -section. In particular, any minimal nondense A -section iselementary. Proof . To prove the “only if” part suppose that A is not dense. If it is notquasidense, then it contains a rank 2 section S of composite order (Theorem 5.5).Since S is elementary nondense, we are done. Suppose that A is quasidesnse and S = V /K is a minimal nondense A -section. Then there exists a non- A -group H ≤ G such that K < H < V.
The minimality of S implies that K is a maximal A -group inside H . So after de-creasing H (if necessary) without loss of generality we can assume that the number p = | H/K | is prime. Next, by the quasidensity of A V/K there exists an A -group M such that K ≤ M ≤ V and the number q = | M/K | is prime. Since H is not an A -group, it follows that p = q and hence H ∩ M = K. GFED@ABC UV ❋❋❋❋❋❋❋①①①①①①① GFED@ABC U ❋❋❋❋❋❋❋ GFED@ABC V q ①①①①①①① GFED@ABC H ❋❋❋❋❋❋❋ p ①①①①①①① GFED@ABC K ❋❋❋❋❋❋❋ GFED@ABC L ①①①①①①① GFED@ABC K ∩ L Figure 1.
Moreover, by the minimality of S , the section V /M is dense. Therefore
M H is an A -group. So M H = V by the minimality of S . Thus | V /K | = pq . To complete theproof it suffices to note that a quasidence but not dense S-ring over a cyclic groupof order pq has to be a proper wreath product (Theorem 5.2).In what follows A is a quasidense S-ring over a cyclic group G . Suppose that S = V /K is an elementary nondense A -section. Denote by H the unique A -subgroup of G such that K < H < V . Then A S = A H/K ≀ A
V/H . The largest A -section which is projectively equivalent to V /H , is obviously of theform
U V /U for some A -group U = U ( S ); similarly, the smallest A -section whichis projectively equivalent to H/K , is of the form L/ ( K ∩ L ) for some A -group L = L ( S ). (The existence of the largest and smallest sections follows from Theo-rem 5.4.) Clearly,(19) 1 < L ≤ U < G.
The relevant part of the lattice of A -groups is given in Fig. 1.The definition of the section U/L associated with S is uniform in the followingsense. Let S be an A -section that contains S as a subsection. Since the mapping ρ G,S defined in Subsection 4.1 induces a lattice epimorphism from G ( A ) to G ( A S ),we have(20) ρ G,S ( U/L ) = U S /L S where U S /L S is the section of the S-ring A S defined by S in the same way as thesection U/L in the S-ring A . In particular, U S /L S = H/H .The following two statements will be used to prove the main result of this section(Theorem 6.5).
Lemma 6.3.
In the above notation the following statements hold: (1) the section S is a quasisubsection of any A -group M U , (2) the section S is a quasisubsection of any A -section G/N with N L . Generalized wreath products arising in nonquasidense case had been studied in [9].
OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 19
Proof . From Lemma 6.1 it follows that c S is an elementary nondense sectionof the S-ring b A . Therefore statement (2) follows from statement (1) by duality.To prove statement (1) we claim that U V /U is a A -quasisubsection of M . Indeed,since U ≤ U M ∩ U V ≤ U V and the number | U V /U | = | V /H | is prime, we have U M ∩ U V ∈ {
U, U V } . However,
U M ∩ U V = U , because otherwise U V M/U M is obviously a multiple ofthe section
U V /U , which contradicts the maximality of it. Therefore
U M ∩ U V = U V , and hence
U V ≤ U M . But U ( M ∩ U V ) =
U M ∩ U V because the lattice G ( A )is modular. Then U ( M ∩ U V ) =
U M ∩ U V = U V.
Since also obviously U ∩ M ∩ U V = U ∩ M , we conclude that the section U V /U isa multiple of ( M ∩ U V ) / ( M ∩ U ). This proves the claim because the latter is an A M -section.Set V ′ /H ′ to be the smallest A -section which is projectively equivalent to V /H ∼ U V /U . Then by the claim in the previous paragraph, V ′ /H ′ is an A M -section. Tocomplete the proof it suffices to verify that V ′ /K ′ ∼ V /K where K ′ = K ∩ V ′ . Let us prove that V /K is a multiple of V ′ /K ′ . Suppose onthe contrary that KV ′ = V . Then KV ′ ≤ H because K ≤ KV ′ ≤ V and H is theonly A -group strictly between K and V . Therefore V ′ ≤ H . On the other hand, V = V ′ H because V /H is a multiple of V ′ /H ′ . Thus V = V ′ H ≤ H which isimpossible. Corollary 6.4.
The section S is a quasisubsection of any A -section M/N with M U and L N . Proof . By statement (1) of Lemma 6.3 the S-ring A M contains a section S ′ = V ′ /K ′ projectively equivalent to S = V /K . Since the section S ′ is elementarynondense (Lemma 6.1), there is a unique A -group H ′ strictly between K ′ and V ′ .From the choice of the group L it follows that H ′ /K ′ ∼ H/K ∼ L/K ∩ L, Moreover, L ≤ H ′ ≤ M . Since also N ≤ M , the hypothesis of Lemma 6.3 is satisfiedfor G = M and S = S ′ . So by statement (2) of this lemma we conclude that thesection M/N has a subsection projectively equivalent to S ′ ∼ S as required.From the Leung-Man theory it follows that any quasidense circulant S-ring thatis not dense, is a proper generalized wreath product (see [4, Theorem 5.3]). Thefollowing theorem gives an explicit form of such a product. Theorem 6.5.
Let A be a quasidense circulant S-ring and S an elementary non-dense section. Then A is a proper U/L -wreath product with U = U ( S ) and L = L ( S ) . Moreover, any dense A -section is either an A U -section or A G/L -section.
Proof . To prove the first statement it suffices to verify by (19) thatrad( X ) ≥ L for all X ∈ S ( A ) G \ U . Suppose on the contrary that this is not true for some X . Then the hypothesis ofCorollary 6.4 holds for M = h X i and N = rad( X ). Therefore S is a quasisubsec-tion of M/N . Since S is elementary nondense, this implies that the S-ring A M/N is not dense. However, this is impossible because A M/N is a quasidense S-ring withtrivial radical (Theorem 5.7).To prove the second statement suppose on the contrary that there exists a dense A -section M/N which is neither an A U - nor A G/L -section. Then S is a qua-sisubsection of M/N by Corollary 6.4. Therefore the S-ring A M/N is not dense.Contradiction.The following auxiliary statement will be used in Section 10.
Lemma 6.6.
Let A ′ be an extension of a quasidense circulant S-ring A . Then G ( A ′ ) = G ( A ) if and only if any elementary nondense section of A is an elementarynondense section of A ′ . Proof . The “only if” part is obvious. To prove the “if” part suppose on thecontrary that there exists a group H ∈ G ( A ′ ) \ G ( A ). Without loss of generalitywe assume that the cyclic group G underlying A and A ′ is minimal possible. Then G = 1. Moreover,(21) U ∈ G ( A ) & U = G = ⇒ U ∩ H = 1and(22) L ∈ G ( A ) & L = 1 = ⇒ LH = G. Indeed, relation (21) follows from relation (22) by duality. To prove (22) suppose onthe contrary that LH = G . Since L and H are A ′ -groups, so is the group LH . Bythe minimality of G we have G ( A G/L ) = G ( A ′ G/L ), and hence LH is an A -group.Again by the minimality this implies that G ( A ′ LH ) = G ( A LH ). It follows that H isan A -group. Contradiction.By the quasidensity of A there exist A -groups U and L such that the numbers | G/U | and | L | are prime. Since 1 < H < G , from (21) and (22) it follows that G = U × H = L × H . Thus the numbers | H | and | G/H | are prime. Therefore theS-ring A is elementary nondense. By the lemma hypothesis this implies that so is A ′ . This implies that H
6∈ G ( A ′ ) in contrast to the choice of H .7. Lifting: dense case
In this section we are to get an analog of the theory developed in Section 6 butthis time for dense S-rings. In what follows under a p -section we mean a sectionwhich is a p -group. Theorem 7.1.
Let A be a circulant dense S-ring. Then rad( A ) = 1 if and only if A contains a non-quasisubprinciple p -section. Proof . The ”if” part is obvious. To prove the ”only if” part suppose thatrad( A ) = 1. Then by [4, Theorem 5.4] the S-ring A is a U/L -wreath product suchthat(23) | G/U | = | L | = p where p is a prime and G is the underlying group of A . To complete the proof itsuffices to verify that the A -section G p is non-quasisubprinciple. Suppose on the OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 21 contrary that G p is a quasisubsection of a principal A -section T = H/K . Then | T p | = | G p | and hence(24) K p = ( G/H ) p = 1 . On the other hand, since A is the U/L -wreath product, Theorem 4.3 implies thatthe S-ring A T is a U T /L T -wreath product where U T /L T = ρ G,T ( U/L ) = ( U ∩ H ) K/ ( L ∩ H ) K. Using (23) and (24) we obtain by comparing p -parts that ( U ∩ H ) K < H and( L ∩ H ) K > K . It follows that U T = T and L T = 1. Therefore A T is a propergeneralized wreath product. However, this is impossible because the section T isprinciple and hence rad( A T ) = 1.As the following example shows, not every minimal non-quasisubprinciple sectionis a p -section. Example.
Let p, q, r be distinct primes such that r − p and q . Let us define S-rings A and A over the group Z rp q as follows A = Cyc( K , , Z rp ) ⊗ Cyc( K , , Z q ) , A = Cyc( K , , Z p ) ⊗ Cyc( K , , Z rq )where K , , K , , K , and K , are groups of order p , q , p and q respectively.The groups K , and K , are uniquely determined; choose the groups K , and K , so that none of the coordinate projections is trivial. Then Cyc( K , , Z rp ) andCyc( K , , Z rq ) are S-rings with trivial radicals. Moreover, using natural identifi-cations we have( A ) p q = Cyc( K , , Z p ) ⊗ Cyc( K , , Z q ) = ( A ) p q where ( A ) p q is the restriction of A to the factorgroup of order p q whereas( A ) p q is the restiction of A to the subgroup of order p q . Then one can formthe generalized wreath product A = A ≀ U/L A over the group Z p q r where U and L are the subgroups of this group of indexand order r respectively. It is not diffcult to verify that U/L is a minimal non-quasisubprinciple A -section.Now let A be a quasidense S-ring over a cyclic group G and S an A -section.Suppose that rad( A S ) > S is a p -section where p is a prime divisor of | G | .Then the S-ring A S is the U /L -wreath product where U and L are subgroupsof S of index and order p respectively. Next, the sets { U ∈ G ( A ) : ρ G,S ( U ) = U } and { L ∈ G ( A ) : ρ G,S ( L ) = L } are nonepmpty, because they contain the groups π − ( U ) and π − ( L ) respectively,where π = π S . Moreover, since the lattice G ( A ) is distributive [16, p.11], these setshave the greatest and least elements. Denote them by U = U ( S ) and L = L ( S ),respectively. Clearly, U ≥ L .The definition of the section U/L associated with S is uniform. Namely, let S be an A -section that contains S as a subsection. Since the mapping ρ G,S definedin Subsection 4.1 induces a lattice epimorphism from G ( A ) to G ( A S ), we have(25) ρ G,S ( U/L ) = U S /L S where U S /L S is the section of the S-ring A S defined by S in the same way as thesection U/L in the S-ring A . In particular, U S /L S = U /L . Lemma 7.2.
In the above notation let
M/N ∈ S ( A ) be such that M U and N L . Then S is an A -quasisubsection of M/N . Proof . Let S = V /K . Then by the definition of U and L we have U = ( U ∩ V ) K/K and L = ( L ∩ V ) K/K.
Since S is a cyclic p -group, this implies that | V p : U p | = p = | L p : K p | . On theother hand, from the hypothesis of the lemma it follows by the maximality of U that M p > U p , and by the minimality of L that N p < L p . Thus V p ≤ M p and N p ≤ K p . Therefore V p N p ′ ≤ M p M p ′ ≤ M and K p N p ′ ≥ N p N p ′ ≥ N. Thus T = V p N p ′ /K p N p ′ is a an A M/N -section. Obviously, T is a multiple of V p /K p .Since the latter section is projectively equivalent to S , we are done.From Theorem 5.2 it follows that any circulant S-ring with nontrivial radical, isa proper generalized wreath product. Statement (1) of the following theorem givesan explicit form of such a product in the quasidense case (cf. Theorem 7.1). Theorem 7.3.
Let A be a quasidense S-ring over a cyclic group G . Suppose that S ∈ S ( A ) is a non-quasiprincipal p -section. Then (1) A is a proper U/L -wreath product where U = U ( S ) and L = L ( S ) , (2) if T ∈ S ( A ) is a subsection of neither U nor G/L , then S is a quasisub-section of T . Proof . To prove the first statement it suffices to verify that rad( X ) ≥ L forall X ∈ S ( A ) G \ U . Suppose on the contrary that this is not true for some X .Then S is a quasisubsection of the section M/N where M = h X i and N = rad( X )(Lemma 7.2). However, this contradicts the theorem hypothesis because the section M/N is principal. To prove the second statement suppose that an A -section T = M/N is a subsection of neither U nor G/L . Then M U and N L . Thus therequired statement immediately follows from Lemma 7.2.8. Coset S-rings
Definition and basic properties.
In this section we introduce and studycirculant coset S-rings. In a sense these rings are antipodes of rational circulantS-rings. Indeed, as we will see below (Theorems 8.3 and 8.4) an S-ring is a cosetone if and only if it can be constructed from group rings by tensor and generalizedwreath products, whereas an S-ring is rational if and only if it can be constructedfrom S-rings of rank 2 in the same way (the latter follows from [11, Theorem 1.2]). Definition 8.1.
An S-ring over an abelian group G is a coset one, if any of itsbasic sets is a coset of a subgroup in G . By definition any basic set X of a coset S-ring A is of the form X = xH forsome group H ≤ G and x ∈ X . It is easily seen that H = rad( X ), and hence(26) X = x rad( X ) In fact, the tensor product here is needless.
OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 23 for any x ∈ X . However, rad( X ) is an A -group. Thus, any basic set of A is a cosetof a uniquely determined A -group.The following statement expressing the “radical monotony property” of a cosetcirculant S-ring, willl be used below. Lemma 8.2.
Let A be a circulant coset S-ring. Then given X, Y ∈ S ( A ) , theinclusion Y ⊂ h X i implies rad( Y ) ≤ rad( X ) . Proof . Let
X, Y ∈ S ( A ). Then from (26) it follows that X and Y are cosetsof the groups rad( X ) and rad( Y ) respectively. Therefore the set X π where π = π G/ rad( X ) , is a singleton consisting of a generator of the group S = h X i / rad( X ).This implies that A S = Z S . If Y ≤ h X i , then Y π is a basic set of A S , andhence Y π ⊂ S is also a singleton. It follows that rad( Y ) π ≤ rad( Y ) π = 1. Thusrad( Y ) ≤ rad( X ) as required.The circulant coset S-rings can be characterized in terms of their sections asfollows. Theorem 8.3.
For a circulant S-ring A the following statements are equivalent: (1) A is a coset S-ring, (2) A S = Z S for any principal A -section S , (3) A S = Z S for any A -section S with trivial radical, (4) A S = Z S for any quasisubprincipal A -section S . Proof . Statements (1) and (2) are equivalent: implication (1) ⇒ (2) followsfrom (26) whereas implication (2) ⇒ (1) follows from the definition of principalsection. Next, implication (4) ⇒ (2) is obvious and implication (2) ⇒ (4) is truebecause the equality in statement (4) is preserved under projective equivalence andtaking subsections. Finally, any principal A -section obviously has trivial radical,and any A -section with trivial radical is subprincipal (Lemma 5.3). Thus implica-tions (3) ⇒ (2) and (4) ⇒ (3) hold.Any primitive section S of a dense circulant S-ring A has prime order. Thereforerad( A S ) = 1. Thus if A is a coset S-ring, then A S = Z S for any primitive A -section S (Theorem 8.3). The converse statement is not true, a counterexample is givenby A = Cyc( {± } , Z ). Theorem 8.4.
The class of circulant coset S-rings is closed under restriction to asection and under tensor and generalized wreath products, and consists of quasidenseS-rings.
Proof . Since any quotient epimorphism takes a coset to a coset, and the productof cosets is a coset, the closedeness statement follows from the definitions of ten-sor and generalized wreath products. The quasidensity statement is true becauseany non-quasidense S-ring has a rank 2 section of composite order (Theorem 5.5)whereas by above no coset S-ring can have such a section.The intersection of circulant coset S-rings is not necessarily a coset one: a coun-terexample is given by the S-ring A = A ∩ A over the group Z pq where A = ZZ p ≀ ZZ q and A = ZZ q ≀ ZZ p with p and q distinct primes. One can see that its rank equals 2, and hence it is notcoset. Moreover, A is even not quasidense (Theorem 5.5). The following statementshows that this is a unique obstacle. Theorem 8.5.
The intersection of circulant coset S-rings is a coset one wheneverit contains a quasidense S-ring.
Proof . Let A = A ∩ A where A and A are circulant coset S-rings. Supposethat A contains a quasidense S-ring. Then by Theorem 5.6 we can assume that A is quasidense. By implication (3) ⇒ (1) of Theorem 8.3 it suffices to verifythat A S = Z S for an A -section S with trivial radical. However, since A i ≥ A for i = 1 ,
2, any such section is an A i -section and ( A i ) S ≥ A S . The quasidensity ofthe S-ring A S implies by Theorem 5.7 that rad(( A i ) S ) = 1. Taking into accountthat A i is a coset S-ring, we conclude that ( A i ) S = Z S (implication (1) ⇒ (3) ofTheorem 8.3). Thus by (6) we have A S = ( A ) S ∩ ( A ) S = Z S as required.In what follows any representation of an S-ring A as a proper generalized wreathproduct, will be called a gwr-decomposition of A . Given an A -section S we say thatthe T ′ -decomposition of A S is lifted to a T -decomposition of A if A is the T -wreathproduct and T S = T ′ (cf. Theorem 4.3). Theorem 8.6.
Let S be a section of a circulant coset S-ring A . Then any gwr-decomposition of the S-ring A S can be lifted to a gwr-decomposition of A . Proof . Let A S be a proper V /K -wreath product. To lift it to a gwr-decompositionof A it suffices to consider two cases depending on whether S is a subgroup or quo-tient of G . By duality the latter case follows from the former one by statement (2)of Lemma 4.4 and statement (3) of Theorem 2.3. In the former case V and K arealso subgroups of G . Set L = K and U = h{ X ∈ S ( A ) : h X i ∩ S ≤ V }i . Clearly, U ∈ G ( A ) and U ≥ V ≥ L . On the other hand, if H and H aresubgroups of G such that H ∩ S ≤ V and H ∩ S ≤ V , then by the distributivityof the lattice G ( A ) we have H H ∩ S = ( H ∩ S )( H ∩ S ) ≤ V. This shows that U ∩ S ≤ V , and hence U ∩ S = V . In particular, U = G . Tocomplete the proof let X be a basic set of A outside U . Then by the definitionof U the group H = h X i ∩ S is not a subgroup of V . Since A S is the V /K -wreathproduct, this implies that K ≤ rad( Y ) where Y is a highest basic set of A H .However, then by Lemma 8.2 we have L = K ≤ rad( Y ) ≤ rad( X ) . Thus A is a proper U/L -wreath product. Since also ρ G,S ( U/L ) =
V /K , we aredone.8.2.
Elementary coset S-rings.
From Theorems 8.3 and 8.4 it follows that anycirculant coset S-ring can be constructed from group rings by generalized wreathproducts. In the rest of this section we are interested in the coset S-rings that areobtained in one iteration of the above process. More precisely, by Theorem 2.1given a section T = U/L of a cyclic group G one can form the S-ring(27) Z ( G, T ) = Z U ≀ T Z ( G/L ) OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 25 because any group ring is dense and the restrictions of both Z U and Z ( G/L ) to T equal Z T . It is easily seen that Z ( G, T ) is a coset S-ring over G . Definition 8.7.
Any S-ring of the form (27) is called elementary coset.
Clearly, the group ring Z G is elementary coset (in this case generalized wreathproduct (27) is not proper). It is also easily seen that every basic set of elemen-tary coset S-ring (27) inside U is a singleton whereas the basic sets outside U are L -cosets. Any elementary coset S-ring is schurian (Theorem 3.6), and the automor-phism group of it is the canonical generalized wreath product of U right by ( G/L ) right in the sense of [9]. For association schemes a similar situation was studied in [15].Let A be elementary coset S-ring (27). Given a function t ∈ L G/U and anelement g ∈ G we define a permutation σ t,g ∈ Sym( G ) by(28) σ t,g : x xt ( X ) g, x ∈ G, where X is the U -coset containing x . The set of all these permutations forms agroup with identity σ , where the first 1 in subscript denotes the function takingevery x to 1, and the multiplication satisfying(29) σ t ,g σ t ,g = σ t t ,g g for all t , t ∈ L G/U and g , g ∈ G . Clearly, { σ ,g : g ∈ G } = G right and σ ,g = σ t, for all g ∈ L where t is the function taking every x to g .It follows from [8, Theorem 7.2] that σ t,g ∈ Aut( A ) whenever t ( U ) = g = 1.Since G right ≤ Aut( A ), the permutation σ t,g is an automorphism of A for all t and g . In fact, statement (2) of the theorem below shows that A has no otherautomorphisms. Theorem 8.8.
Let A = Z ( G, T ) be an elementary coset S-ring (27) and S an A -section. Then (1) A S = Z ( S, T S ) where T S is the section defined in (17) , (2) Aut( A ) = { σ t,g : t ∈ L G/U , g ∈ G } , (3) Aut( A ) S = Aut( A S ) . Proof . By Theorem 4.3 the S-ring A S is the T S -wreath product of the S-rings A U S and A S/L S = A ( G/L ) S which are Cayley isomorphic by Lemma 4.1 and The-orem 4.2 to the S-rings A S U and A S G/L respectively. Since A U = Z U and S U isan A U -section, as well as A G/L = Z ( G/L ) and S G/L is an A G/L -section, we have A S U = Z S U and A S G/L = Z ( S G/L ) which proves statement (1).To prove statement (2) denote by Γ the group in the right-hand side of theequality, Then from the discussion before the theorem it follows that Γ ≤ Aut( A ).Therefore to check the reverse inclusion it suffices to prove that | Aut( A ) | ≤ | Γ | .However, since A G/L = Z ( G/L ), we haveAut( A ) G/L = (
G/L ) right = Γ G/L . To complete the proof we show that the kernel of the epimorphism π : Aut( A ) → Aut( A ) G/L is contained in Γ, more precisely that any σ ∈ ker( π ) is of the form σ t, for some t ∈ L G/U . Note that such a permutaton σ leaves each X ∈ G/U fixed as a set.Therefore all we need to prove is that σ X acts on X by multiplying by an element l = l ( X ) belonging to L . Let g X ∈ G U → X . Then obviously the permutation σ ′ = g X σg − X belongs to ker( π ) and leaves the set U fixed. Taking into accountthat A U = Z U , we see that ( σ ′ ) U ∈ Aut( A ) U = U right . Therefore since σ ′ leavesalso any L -coset fixed, there exists l ∈ L for which u g X σg − X = u σ ′ = lu, u ∈ U. When u runs over U , the element x = u g X runs over X . Therefore the aboveequality implies that x σ = lx for all x ∈ X , as required.To prove statement (3) we observe that obviously Aut( A ) S ≤ Aut( A S ). To verifythe converse inclusion, let f ′ ∈ Aut( A S ) and S = H/K . Then by statements (1)and (2) we have f ′ = σ t ′ ,g ′ where t ′ ∈ L S/U S S and g ′ ∈ S . However, from (29) itfollows that σ t ′ ,g ′ = σ t ′ , σ ,g ′ . Moreover, the permutation σ ,g ′ ∈ S right can belifted to a permutation in G right , because obviously Γ S = S right where Γ is thesetwise stabilizer of H in G right . Thus we can assume that f ′ = σ t ′ , .To verify that f ′ can be lifted to a permutation in Aut( A ) it suffices to checkthat there exists t ∈ L G/U such that(30) ( σ t, ) S = σ t ′ , . To define the function t given X ∈ G/U set t ( X ) = 1 if the set X ∩ H is empty.Suppose that it is not empty. Then X ∩ H is a coset of H ∩ U . Recall that U S = ( U ∩ H ) K/K and L S = ( L ∩ H ) K/K.
Therefore, the first equality implies that X ′ := π ( X ∩ H ) belongs to S/U S where π = π S . Moreover, the set L ∩ π − ( t ′ ( X ′ ) is not empty by the second equality. Nowset t ( X ) to be any element of that set. It follows that in any case t ∈ L G/U , andthe permutation σ t, leaves H fixed. Thus, given x ′ ∈ S we have( x ′ ) σ St, = x ′ π ( t ( X )) = x ′ t ′ ( X ′ ) = ( x ′ ) σ t ′ , where X ′ is the U S -coset containing x ′ , which proves (30).In general, an extension of a coset S-ring is not necessary coset: a non-coset S-ringCyc( {± } , Z ) contains a coset S-ring Cyc( K, Z ) where K = Aut( Z ). However,we can prove the following auxiliary statement to be used in Section 9. Lemma 8.9.
Let A be an elementary coset S-ring (27) . Suppose that either | L | or | G/U | is prime. Then any extension of A is an elementary coset S-ring. Proof . By duality (Theorem 2.3) without loss of generality we can assume that | L | is prime. Let A ′ ≥ A . Denote by U ′ the group consisting of all x ∈ G for which { x } is a basic set of A ′ . Clearly, U ′ is an A ′ -group containing U . We claim that anybasic set not in U ′ , is a basic set of A ′ . Then since A ′ ≥ A , we have A ′ = Z ( G, T ′ )where T ′ = U ′ /L , as required.Suppose that the claim is not true. Then there exists a basic set X U ′ of A ,that is not a basic set of A ′ . It follows that X ⊂ G \ U . Since A = Z ( T, G ), wehave(31) X = xL OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 27 for all x ∈ X . Moreover, X is the union of at least two basic sets of A ′ . If oneof them, say X ′ , has nontrivial radical, then the latter coincides with L , because A ′ G/L ≥ A
G/L = Z ( G/L ). But then from (31) it follows that | L | = | X | ≥ | X ′ | ≥ | L | . Therefore X = X ′ . Contradiction.Now we have rad( X ′ ) = 1 for all X ′ ∈ S ( A ′ ) X . Since the group H := h X i iscyclic, there exists such an X ′ such that h X ′ i = H . Therefore rad( A ′ H ) = 1. Onthe other hand, since A H ≤ A ′ H and A is quasidence, we conclude by Theorem 5.6,then A ′ H is also quasidence. Thus by Theorem 5.7 we have(32) A ′ H = Cyc( K, H )for some group K ≤ Aut( H ). Clearly, K is a subgroup of the stabilizer of 1 inAut( A H ). Since this stabilizer is a p -group (statements (1) and (2) of Theorem 8.8)where p = | L | , it follows that K is also a p -group. This implies that the cardinalityof every K -orbit is a p th power. Since X is a K -invariant set of cardinality p ,equality (32) implies that either every element in S ( A ′ ) X is singleton, or S ( A ′ ) X = { X } . In both cases we come to a contradiction with the choice of X .9. Schurity and separability of coset S-rings
The theory of coset circulant S-rings developed in Section 8 enables us to provethe following theorem which is the main result of the section.
Theorem 9.1.
Any circulant coset S-ring is schurian and separable.
We will deduce Theorem 9.1 in the end of the section from two following auxiliarystatements on the automorphism group of a coset S-ring. In what follows, if an S-ring A is the S -wreath product where S ∈ S ( A ), then we say that S is a gwr-section of A . Theorem 9.2.
Let A be a coset S-ring over a cyclic group G . Then the group Aut( A ) is generated by the automorphism groups of elementary coset S-rings Z ( G, S ) where S runs over all gwr-sections of A . Proof . Induction on the order of G . The base case of the induction is obvious.Let G = 1 and f ∈ Aut( A ). By Theorem 8.4 the S-ring A is quasidense. Sothere exists an A -group L of prime order. Set S = G/L . Since f S ∈ Aut( A S ), bythe inductive hypothesis f S is the product of automorphisms of elementary cosetS-rings Z ( S, T ′ ) where T ′ runs over all gwr-sections of A S . On the other hand, byTheorem 8.6 any T ′ -decomposition of A S can be lifted to a T -decomposition of A .Set B = Z ( G, T ). Then B S = Z ( S, T ′ ) by statement (1) of Theorem 8.8. Moreover,by statement (3) of that theorem we also have Aut( B ) S = Aut( B S ). Thus to write f as the product of automorphisms of elementary coset S-rings, without loss ofgenerality we can assume that(33) f S = id S . Denote by A ′ the S-ring over G associated with the group Γ = h G right , f i . Inparticular, f ∈ Aut( A ′ ). Due to (33) we have Γ S = S right . Therefore(34) A ′ G/L = Z ( G/L ) . Moreover, A ′ ≥ A because f ∈ Aut( A ). It follows that A ′ L ≥ A L . On the otherhand, A L is a coset S-ring (Theorem 8.4) with trivial radical because | L | is prime.Thus A L = Z L , and hence A ′ L = Z L . Together with (34) this implies that A ′ isan extension of elementary coset S-ring Z ( G, L/L ). Therefore by Lemma 8.9 weconclude that(35) A ′ = Z ( G, T ′ )where T ′ = U ′ /L with U ′ being the largest A ′ -group for which A ′ U ′ = Z U ′ .Denote by U the largest A -subgroup of G inside U ′ . Clearly, L ≤ U . Take aset X ∈ S ( A ) outside U . Then by the definition of U there exists a basic A ′ -set X ′ ⊂ X that is outside U ′ . By (35) this implies that X ′ is an L -coset. However, X is a coset of rad( X ) because A is a coset S-ring. Thus L ≤ rad( X ). This provesthat T := U/L is a gwr-section of A . Besides, since U ≤ U ′ we have Z ( G, T ) ≤ Z ( G, T ′ ) . Since f ∈ Aut( A ′ ), this inclusion together with (35) implies that f is an automor-phism of the elementary coset S-ring Z ( G, T ), as required.
Theorem 9.3.
Let A be a circulant coset S-ring. Then Aut( A ) S = Aut( A S ) forany A -section S . Proof . Obviously, Aut( A ) S ≤ Aut( A S ). To prove the reverse inclusion weobserve that by Theorem 9.2 each automorphism of A S can be written as theproduct of automorphisms of elementary coset S-rings Z ( S, T ′ ) where T ′ runs overall gwr-sections of A S . On the other hand, by Theorem 8.6 any T ′ -decompositionof A S can be lifted to a T -decomposition of A . Thus the required statement followsfrom statement (3) of Theorem 8.8. Proof of Theorem 9.1.
Let A be a coset S-ring over a cyclic group G . Wewill prove both statements by induction on the order of this group. Let | G | > A ) = 1, then A = Z G by implication (1) ⇒ (3) of Theorem 8.3 and thestatements are obvious. Let now rad( A ) >
1. Then by statement (1) of Theorem 5.2there is a gwr-decomposition(36) A = A U ≀ U/L A G/L . By Theorem 8.4 the S-rings A U and A G/L are coset ones. Thus they are schurianand separable by the inductive hypothesis.To prove that A is schurian set ∆ = Aut( A U ) and ∆ = Aut( A G/L ). Thenby Theorem 3.6 it suffices to verify that (∆ ) U/L = (∆ ) U/L . However, fromTheorem 9.3 it follows that(∆ ) U/L = Aut( A U ) U/L = Aut( A U/L ) = Aut( A G/L ) U/L = (∆ ) U/L as required.To prove that A is separable let ϕ : A → A ′ be a similarity. Then by state-ment (1) of Theorem 3.3 the S-ring A ′ is the U ′ /L ′ -wreath product where U ′ = U ϕ and L ′ = L ϕ . By the inductive hypothesis the similarities ϕ U : A U → A ′ U ′ and ϕ G/L : A G/L → A ′ G ′ /L ′ OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 29 are induced by some bijections, say f and f . Given X ∈ G/U let us fix two bijec-tions g ∈ G U → X and g ′ ∈ G ′ X ′ → U ′ . By Lemma 3.4 the bijection g U/L ( f ) X/L ( g ′ ) X ′ /L ′ induces the similarity ϕ U/L . It follows that(37) g U/L ( f ) X/L ( g ′ ) X ′ /L ′ (( f ) U/L ) − ∈ Aut( A U/L ) . By Theorem 9.3 there exists an automorphism h X ∈ Aut( A U ) such that its restric-tion to U/L coincides with the automorphism in the left-hand side of (37). Now,the family of bijections f X := g − h X f ( g ′ ) − , X ∈ G/U, defines a bijection f : G → G ′ such that ( G/U ) f = G ′ /U ′ and ( G/L ) f = G ′ /L ′ .Then obviously(38) gf X g ′ = gf X g ′ = h X f . By (12) this implies that the bijection gf X g ′ induces the similarity ϕ U for all X .This proves the second part of condition (13). Let us check the first part of thatcondition. For any X ∈ G/U from the definition of h X and (38) it follows that f X/L = ( g − h X f ( g ′ ) − ) X/L =( g U/L ) − g U/L ( f ) X/L ( g ′ ) X ′ /L ′ (( f ) U/L ) − ( f ) U/L (( g ′ ) X ′ /L ′ ) − = ( f ) X/L . Therefore f G/L = f , and hence the bijection f G/L induces the similarity ϕ G/L .Thus by Theorem 3.5 the bijection f induces the similarity ϕ as required.The following statement can in a sense be regarded as a combinatorial analog ofTheorem 9.1. Corollary 9.4.
Any circulant coset S-ring is an intersection of elementary cosetS-rings over the same group.
Proof . Let A be a circulant coset S-ring. Denote by A ′ the intersection of allelementary coset S-rings Z ( G, S ) where S runs over all gwr-sections of A . Since A is quasidense (Theorem 8.4), the S-ring A ′ is coset (Theorem 8.5). So fromTheorem 9.1 it follows that it is schurian. Therefore A ′ equals the S-ring associatedwith the group Γ generated by the automorphism groups of the above S-rings Z ( G, S ). On the other hand, by Theorem 9.1 the S-ring A is also schurian. ByTheorem 9.2 this implies that A equals the S-ring associated with the group Γ.Thus A = A ′ . 10. Coset closure
Relative coset closure.
We start with developing a technique to find theschurian closure of a quasidense circulant S-ring A (Theorem 1.2). The key pointof this technique is its coset closure A defined in Introduction (Definition 1.1) asthe intersection of all coset S-rings containing A . By Theorem 8.5 the S-ring A isa coset one. Moreover,(39) ( A ) S = Z S, S ∈ S ( A )where S ( A ) is the class of all quasisubprincipal A -sections. Indeed, by Theo-rem 4.2 this is reduced to the case of a principal S , in which the required statementfollows from Theorem 5.7 and implication (1) ⇒ (3) of Theorem 8.3. Thus theS-ring A equals the intersection of all coset S-rings A ′ such that(40) A ′ ≥ A and ( A ′ ) S = Z S for all S ∈ S ( A ). For induction reasons it is convenient to generalize the cosetclosure concept by permitting the section S to run over a larger class S . Definition 10.1.
A class S ⊂ S ( A ) is admissible with respect to A (or A -admissible), if S prin ( A ) ⊂ S ⊂ S cyc ( A ) and S is closed under taking A -quasisub-sections. Given an A -admissible class S and an A -section S we set S S = ρ G,S ( S ) wherethe mapping ρ G,S is defined in Subsection 4.1. Since ρ G,S is identical on the set S ( S ) ⊂ S ( G ), we have(41) S S = { ρ G,S ( S ′ ) : S ′ ∈ S , S ′ (cid:22) S } . The following statement gives the properties of admissible classes.
Lemma 10.2.
Let S be an admissible class with respect to a quasidense circulantS-ring A . Then (1) S ( A ) ⊂ S , (2) the class S S is A S -admissible for any S ∈ S ( A ) , (3) if S ∈ S ( A ) \ S , then A S admits a gwr-decomposition. Proof . Statements (1) is obvious. Statement (2) follows from (41). To proveStatement (3) let S ∈ S ( A ) \ S . Then rad( A S ) = 1, for otherwise S is a subsectionof a principal A -section (Lemma 5.3) whereas S contains all such sections. Thusthe required statement follows from Theorem 5.2.Let A be a quasidence circulant S-ring and S an A -admissible class. Then thereis at least one coset S-ring A ′ , namely the group ring, for which relations (40) holdfor all S ∈ S . This justifies the following definition. Definition 10.3.
The coset closure A , S of A with respect to S is the intersectionof all coset S-rings A ′ such that relations (40) hold for all S ∈ S . Clearly, A , S ≥ A . Moreover, from (6) it follows that ( A , S ) S = Z S for all S ∈ S . Besides, the discussion in the first paragraph of the section shows that(42) A = A , S when S = S ( A ) . The sense of the following definition will be clarified in Corollary 10.10. Withany A -admissible class S we associate a larger class of sections defined as follows b S = { S ∈ S ( A ) : S p ∈ S for all primes p dividing | S |} . The class b S is closed under taking quasisubsections, but generally is not admissiblebecause may contain non-cyclotomic sections. As the example on page 21 shows,in general it can be larger than S . Lemma 10.4.
Let A be a quasidence S-ring over a cyclic group G . Then for any A -admissible class S we have (1) the S-ring A , S is a coset one, (2) ( A , S ) S = Z S for all S ∈ S , (3) any coset belonging to a section in b S is an A , S -set. Proof . Statement (1) immediately follows from Theorem 8.5. Statement (2)follows from the remark after the definition of the coset closure A , S . To provestatement (3) let A ′ be a coset S-ring such that relations (40) hold for all S ∈ S . OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 31
Then given a section S ∈ b S we have S p ∈ S , and hence ( A ′ ) S p = Z S p for all primes p dividing | S | . This implies that ( A ′ ) S = Z S . It follows that if S = U/L , then any L -coset in U is an A ′ -set. Since A is the intersection of all such A ′ , it is an A -setas required.10.2. Lifting.
Let S be an admissible class with respect to a quasidence S-ring A over a cyclic group G . Suppose that A is the S -wreath product where S = U/L isan A -section. We say that this product is S -consistent if any section in S is eitheran A U - or A G/L -section. Below to simplify notation we write ( A S ) , S instead of( A S ) , S S . Theorem 10.5.
Let A be a circulant quasidense S-ring, S an A -admissible classand S an A -section. Then the following conditions are equivalent: (1) ( A S ) , S = Z S , (2) S b S , (3) there exists an S S -consistent gwr-decomposition of A S , (4) there exists an S S -consistent gwr-decomposition of A S that can be lifted toan S -consistent gwr-decomposition of A . Proof . Let us prove implication (1) ⇒ (2). Suppose on the contrary that S ∈ b S .Then obviously S ∈ c S S . By statement (3) of Lemma 10.4 applied to A = A S , S = S S and the section S/
1, this implies that ( A S ) , S = Z S . Contradiction.To prove implication (2) ⇒ (4) let S b S . Then there is a prime divisor p of | S | such that S p S . Suppose first that S p is not an A S -group. Then the S-rings A S and A are not dense. Then by Theorem 6.2 there exists an elementary nondense A S -section S . By the first part of Theorem 6.5 applied to the S-rings A and A S ,and equality (20) there are gwr-decompositions(43) A = A U ≀ U/L A G/L and A S = A U S ≀ U S /L S A S/L S where U = U ( S ) and L = L ( S ) and U S /L S = ρ G,S ( U/L ). It follows thatthe first one is a lifting of the second. Finally, the second part of Theorem 6.5together with the fact that any section in the class S is dense, shows that thesegwr-decompositions are S - and S S -consistent respectively. Thus statement (4)holds in this case.Let now S p be an A S -group. Then by Lemma 10.2 the hypothesis of Theorem 7.3holds for the S-rings A and A S with S = S p in both cases. So by statement (1) ofthis theorem there are gwr-decompositions (43) and due to (25) the first one is alifting of the second. To prove that the first decomposition is S -consistent, supposeon the contrary that there exists T ∈ S which is neither A U - nor A G/L -section.Then by statement (2) of that theorem S p is a quasisubsection of T . Therefore S p ∈ S , which contradicts the assumption on S p . The S -consistency of the seconddecomposition is proved similarly. Thus statement (4) holds in this case too.Implication (4) ⇒ (3) is obvious. To prove implication (3) ⇒ (1) without lossof generality we can assume that S = G . Suppose that the S-ring A admits an S -consistent U/L -decomposition. By Theorem 2.1 we can form the S-ring B = A ′ U ≀ U/L A ′ G/L where A ′ = A , S . By Theorem 8.4 this S-ring is a coset one. Moreover, theconsistency property implies that any section T ∈ S is either an A U - or A G/L -section, and hence either an A ′ U - or A ′ G/L -section. Therefore B T = Z T for allsuch T . Thus, by the definition of the coset closure we have B ≥ A ′ . So from theminimality property of the generalized wreath product it follows that the S-ring A ′ admits the U/L -decomposition. Therefore A ′ = Z G as required.The following auxiliary statement will be used in proving the theorems in thenext subsection. Lemma 10.6.
Let A be a circulant quasidense S-ring, S an A -admissible classand S an A -section. Suppose that A S admits an S S -consistent T -decompositionthat can be lifted to an S -consistent gwr-decomposition of A . Then ( A , S ) S admitsthe T -decomposition. Proof . By Theorem 4.3 without loss of generality we can assume that S = G .Then we have to verify A , S admits the T -decomposition whenever A admits an S -consistent T -decomposition. However, the latter implies that any T ′ ∈ S iseither an A U - or A G/L -section where
U/L = T . It follows that T ′ is either A ′ U - or A ′ G/L -section where A ′ is the elementary coset S-ring Z ( G, T ). Therefore( A ′ ) T ′ = Z T ′ , T ′ ∈ S . So by the definition of the coset closure we have A ′ ≥ A , S . Since obviously T isan A , S -section, the S-ring A , S is the T -wreath product, as required.10.3. Main properties.
Here we study the coset closure in detail and, in partic-ular, find its explicit structure.
Theorem 10.7.
Let A be a quasidense circulant S-ring. Then G ( A ) = G ( A , S ) for any A -admissible class S . Proof . Below to simplify notations we omit the letter S in subscript. Thetheorem statement is obviously true when the S-ring A is dense. Suppose that itis not dense. Then by Theorems 6.2 and 6.5 the S-ring A admits an S -consistent U/L -decomposition.
Lemma 10.8.
Let B be a circulant S-ring over G . Suppose that B is a U/L -wreathproduct. Then G ( B ) = G ( B U ) ∪ π − ( G ( B G/L )) where π = π G/L is the quotient epimorphism from G to G/L . Proof . Obviously, the right-hand side is contained in the left-hand one. Con-versely, let H ∈ G ( B ). Without loss of generality we can assume that H U . Thenany highest basic set of B H is outside U . Since B is a U/L -wreath product, theradical of that set contains L . Thus L ≤ H as required.By Lemma 10.8 and Lemma 10.6 with S = G this implies that G ( A ) = G ( A U ) ∪ π − ( G ( A G/L )) and G ( A ) = G (( A ) U ) ∪ π − ( G (( A ) G/L )) . On the other hand, by induction we have(44) G ( A U ) = G (( A U ) ) and G ( A G/L ) = G (( A G/L ) ) . Thus it suffices to verify that given S ∈ { U, G/L } we have(45) G (( A S ) ) = G (( A ) S ) . OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 33
By Lemma 6.6 with A = ( A S ) and A ′ = ( A ) S all we need to prove is that anyelementary nondense section of ( A S ) is an elementary nondense section of ( A ) S .However by (44), any such section T is an A S -section. Therefore T is an ele-mentary nondense section of A . In particular, A T admits a unique S -consistentgwr-decomposition, and by Theorem 6.5 this decomposition can be lifted to an S -consistent gwr-decomposition of A . By Lemma 10.6 this implies that T is anelementary nondense section of A , and hence of ( A ) S .It is easily seen that ( A S ) , S ≤ ( A , S ) S for all A -sections S . The followingtheorem refines this simple statement. Theorem 10.9.
Let A be a quasidense circulant S-ring and S an A -admissibleclass. Then ( A S ) , S = ( A , S ) S for any A -section S . Proof . Below to simplify notations we omit the letter S in subscript. Supposeon the contrary that ( A S ) < ( A ) S for some A -section S . Then there exist basicsets X and Y of the S-rings ( A S ) and ( A ) S respectively, such that Y is a propersubset of X . From Theorem 10.7 it follows that G (( A ) S ) = G ( A S ) = G (( A S ) ) . Therefore h Y i is an ( A S ) -group. However, the set X ∩ h Y i is not empty. Thus, X ⊂ h Y i . On the other hand, X and Y are cosets, because the S-rings ( A S ) and ( A ) S are coset ones. Since Y is a proper subset of X , this implies thatrad( Y ) < rad( X ). Thus, (( A S ) ) T = Z T where T = h Y i / rad( Y ). This impliesthat ( A T ) = (( A S ) T ) ≤ (( A S ) ) T < Z T. By implication (1) ⇒ (4) of Theorem 10.5 the hypothesis of Lemma 10.6 is satisfiedfor S = T . Therefore the S-ring ( A ) T admits a gwr-decomposition. It follows thatrad(( A ) T ) = 1, which is impossible because T is a principal section of the S-ring( A ) S . Corollary 10.10.
In the conditions of Theorem 10.9 the following statements hold: (1) b S = S ( A , S ) , (2) b S is an A -admissible class if and only if b S ⊂ S cyc ( A ) , (3) S = S ( A ) = \ S ( A ) where S and A are as in Introduction. Proof . Statement (1) immediately follows from equivalence (1) ⇔ (2) of Theo-rem 10.5 and Theorems 10.9 and 8.3. Statement (2) follows from statement (1) andTheorem 10.7. Statement (3) is a special case of statement (1) for S = S ( A ).Let A be a quasidense circulant S-ring and S an A -admissible class. For a basicset X of A set(46) L S ( X ) = \ h X i /L ∈ b S ,L ≤ rad( X ) L. Certainly, at least one group L does exist because b S ⊃ S prin ( A ), and hence onecan take L = rad( X ). It should be mentioned that since the class b S is closed withrespect to taking subsections, the left-hand side of (46) does not change when theintersection is taken over all sections U/L ∈ b S having h X i / rad( X ) as a subsection.Clearly, L S ( X ) is an A -group contained in rad( X ). Therefore X is a union ofcosets of it; the set of all of them is denoted by X/L S ( X ). Theorem 10.11.
Let A be a quasidense circulant S-ring and S an A -admissibleclass. Then (1) S ( A , S ) = S S ∈S ( A ) X/L S ( X ) = { xL S ( X ) : x ∈ X ∈ S ( A ) } , (2) elements x and y of a basic set X of A are in the same basic set of A , S ifand only if π S ( x ) = π S ( y ) for any section S ∈ b S having h X i / rad( X ) as asubsection. Proof . Statement (2) immediately follows from statement (1) and the remearkbefore the theorem. To prove statement (1) let X ∈ S ( A ) and x ∈ X . Denoteby X the basic set of the S-ring A , S that contains x . Then X = xL for some A , S -subgroup L , because A , S is a coset S-ring. Moreover, by statement (3) ofLemma 10.4 the set X is contained in some L -coset in U := h X i for any group L such that U/L ∈ b S . By (46) this implies that L ≤ L S ( X ) . If L = L S ( X ), then xL S ( X ) = X and we are done. Suppose that L < L S ( X ).By Theorem 10.7 the group U := h X i is an A -group. Since X intersects U , thisimplies that h X i = U . Moreover, due to the assumption we also have L < rad( X ).On the other hand, since S := U /L is a principal section of a coset S-ring A , S ,Theorem 8.3 implies that ( A , S ) S = Z S . By Theorem 10.9 and implication (2) ⇒ (1)of Theorem 10.5 this implies that S ∈ b S . Since h X i / rad( X ) is a subsection of S ,we obtain that L S ( X ) ≤ L . Contradiction.11. Multipliers
Let A be a quasidense circulant S-ring and S an A -admissibe class. In whatfollows for an A -section S we set Aut A ( S ) = Aut( A S ) ∩ Aut( S ). Definition 11.1.
An element
Σ = { σ S } of the direct product Q S ∈ S Aut A ( S ) ,is called an S -multiplier of A if the following two conditions are satisfied for allsections S , S ∈ S : (M1) if S (cid:23) S , then ( σ S ) S = σ S , (M2) if S ∼ S implies m ( σ S ) = m ( σ S ) .The group of all S -multipliers of A is denoted by Mult S ( A ) . It should be noted that if the class S defined in (3) is contained in S cyc ( A ),then it is admissible (Corollary 10.10) and Mult( A ) = Mult S ( A ). The followinglemma gives a natural way to construct S -multipliers. Lemma 11.2.
Suppose that we are given γ ∈ Aut( A ) such that γ S ∈ Aut A ( S ) for all S ∈ S . Then the family Σ( γ ) = { σ S ( γ ) } S ∈ S where σ S ( γ ) = γ S , is an S -multiplier. Proof . Condition (M1) is obvious. Besides, from Theorem 4.2 it follows that m ( γ S ) = m ( γ S ) for any projectively equivalent sections S , S ∈ S . Therefore m ( σ S ( γ )) = m ( σ S ( γ )). Thus condition (M2) is also satisfied for Σ( γ ), and we aredone.Let Σ ∈ Mult S ( A ) and T ∈ S ( A ). Then S T is an A T -addmissible class bystatement (2) of Lemma 10.2, and S T ⊂ S by (41). Conditions (M1) and (M2)are obviously satisfied for the restriction Σ T of Σ to S T , which is by definition OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 35 considered as an element of the direct product Q S ∈ S T Aut A ( S ). This proves thefollowing statement. Lemma 11.3.
The family Σ T is an S T -multiplier of the S-ring A T . We are going to construct S -multipliers of A by means of similarities belongingto the set Φ . S ( A ) = { ϕ ∈ Φ( A , S ) : X ϕ = X for all X ∈ S ( A ) } . Namely, with any ϕ ∈ Φ , S ( A ) we associate an element Σ ϕ = { σ S } of the group Q S ∈ S Aut A ( S ) where the automorphisms σ S are defined as follows. Let S ∈ S .Then by statement (2) of Lemma 10.4 we have ( A , S ) S = Z S . Therefore thereexists a uniquely determined automorphism σ S ∈ Aut( S ) that induces the restric-tion ϕ S of the similarity ϕ to S (Lemma 3.2). From the choice of ϕ it follows that σ S ∈ Aut A ( S ). Lemma 11.4.
The family Σ ϕ is an S -multiplier of the S-ring A . Moreover, ϕ T ∈ Φ , S T ( A T ) and (Σ ϕ ) T = Σ ϕ T for all T ∈ S ( A ) . Proof . By Theorem 9.1 the similarity ϕ is induced by an isomorphism γ of theS-ring A , S . Clearly, γ can be chosen so that 1 γ = 1. Then by Lemma 3.2 wehave γ S ∈ Aut A ( S ) for all S ∈ S . Besides, γ ∈ Aut( A ) by Lemma 3.7. So fromLemma 11.2 it follows that Σ( γ ) is an S -multiplier of A . Thus, the first statementfollows because Σ ϕ = Σ( γ ).To prove the second statement, let T ∈ S ( A ). Then ϕ T ∈ Φ(( A , S ) T ). ByTheorem 10.9 this implies that ϕ T ∈ Φ(( A T ) , S ). So ϕ T ∈ Φ , S T ( A T ). The restof the statement easily follows from the definitions.To simplify notation we will write Φ , S ( A T ) instead of Φ , S T ( A T ). Theorem 11.5.
Let A be an quasidense circulant S-ring. Then the mapping (47) Φ , S ( A ) → Mult S ( A ) , ϕ Σ ϕ is a group isomorphism for any A -admissible class S . Proof . The mapping (47) is obviously a group homomorphism. To prove itsinjectivity suppose that Σ ϕ = Σ ψ for some ϕ, ψ ∈ Φ , S ( A ). Then obviously ϕ S = ψ S for all S ∈ S . By Lemma 3.2 the equality also holds for all S ∈ b S because anysimilarity of Z S is uniquely determined by its restrictions to the Sylow sections S p .However, b S = S ( A , S ) by Corollary 10.10. Therefore ϕ and ψ are equal on allprincipal A , S -sections. Thus ϕ = ψ by Lemma 3.1.Let us prove the surjectivity of homomorphism (47) by induction on the size ofthe group G underlying A . Let Σ = { σ S } be an S -multiplier of A . First, supposethat A , S = Z G . By implication (1) ⇒ (2) of Theorem 10.5 this implies that G ∈ b S . So any Sylow subgroup of G belongs to S . It follows that A is a subtensorproduct of cyclotomic S-rings. In this case the following statement holds. Lemma 11.6.
There exists γ ∈ Aut A ( G ) such that Σ = Σ( γ ) . Proof . Set γ to be the unique automorphism of G such that γ G p = σ G p for allSylow subgroups G p of G . We claim that(48) σ S ( γ ) = σ S , S ∈ S , where σ S ( γ ) is the S -component of the family Σ( γ ). Indeed, the automorphism σ S is uniquely determined by its p -components ( σ S ) p ∈ Aut( S p ). Besides, S p ∈ S because S ∈ S , and hence ( σ S ) p = σ S p by Definition 11.1. Since G p ∈ S , each σ S p is in its turn uniquely determined by the automorphism σ G p . By the definition of γ this proves (48).To complete the proof of the lemma let us verify that γ ∈ Aut( A ). Since γ ∈ Aut( G ), by Lemma 3.7 for A ′ = Z G , it suffices to check that X γ = X for all basicsets X of A . However, the section S = h X i / rad( X ) belongs to S . Moreover, X isa disjoint union of rad( X )-cosets. Since σ S ∈ Aut( A S ), equality (48) implies thatthe automorphism γ permutes the cosets in this union, as required.By Lemma 11.6 we have Σ = Σ( γ ). On the other hand, it is easily seenthat Σ( γ ) = Σ ϕ where ϕ = ϕ γ is the similarity induced by the automorphism γ (Lemma 3.2). Thus Σ = Σ ϕ as required.Now, assume that A , S = Z G . Then by implication (1) ⇒ (3) of Theorem 10.5for S = G . there exists an S -consistent U/L -decomposition of A . By the inductivehypothesis applied to the S-ring A U and S U -multiplier Σ U , as well to the S-ring A G/L and S G/L -multiplier Σ
G/L , there exist similarities ϕ ∈ Φ , S ( A U ) and ϕ ∈ Φ , S ( A G/L ) such that(49) Σ U = Σ ϕ and Σ G/L = Σ ϕ . Furthermore, ( A U ) , S = ( A , S ) U and ( A G/L ) , S = ( A , S ) G/L by Theorem 10.9.Therefore(50) (( A U ) , S ) U/L = ( A , S ) U/L = (( A G/L ) , S ) U/L . However, again by Theorem 10.9 we have ( A , S ) U/L = ( A U/L ) , S . It follows thatthe similarities ( ϕ ) U/L and ( ϕ ) U/L belong to Φ , S ( A U/L ).By the first equality in (49) and Lemma 11.4 applied to A = A U , ϕ = ϕ and T = U/L , we obtain thatΣ ( ϕ ) U/L = (Σ ϕ ) U/L = (Σ U ) U/L = Σ
U/L . Similarly, by the second equality in (49) and Lemma 11.4 applied to A = A U , ϕ = ϕ and T = U/L we haveΣ ( ϕ ) U/L = (Σ ϕ ) U/L = (Σ
G/L ) U/L = Σ
U/L . Thus Σ ( ϕ ) U/L = Σ ( ϕ ) U/L and by the injectivity statement we have( ϕ ) U/L = ( ϕ ) U/L . On the other hand, by Theorem 2.1 and equality (50) one can form the
U/L -wreath product A ′ of the S-rings ( A U ) , S and ( A G/L ) , S . Thus by statement (2)of Theorem 3.3 there exists a uniquely determined similarity ϕ ∈ Φ( A ′ ) such that(51) ϕ U = ϕ and ϕ G/L = ϕ . However, A is the U/L -wreath product. By Corollary 10.6 so is A , S . Since therestrictions of the latter S-ring to U and G/L coincide with the correspondingrestrictions of the S-ring A ′ (see above), we conclude that A ′ = A , S . Thus ϕ ∈ Φ( A , S ). Since ϕ ∈ Φ , S ( A U ) and ϕ ∈ Φ , S ( A G/L ), we conclude that ϕ ∈ Φ , S ( A ). OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 37
To complete the proof let us verify that Σ = Σ ϕ . To do this we observe thatfrom (49) and (51) it follows that (Σ ϕ ) U = Σ U and (Σ ϕ ) G/L = Σ
G/L . This provesthe required statement because by the S -consistency property, any section in S iseither A U - or A G/L -section.12.
Proof of Theorems 1.2 and 1.3
The following auxiliary statement is interesting by itself. In particular, it showsthat the subgroup lattices of a quasidense S-ring and its schurian closure are equal.It seems that this statement could be generalized to all circulant S-rings.
Lemma 12.1.
Let A be a quasidence cirulant S-ring and A ′ its schurian closure.Then G ( A ) = G ( A ′ ) and A = ( A ′ ) . Proof . From Theorem 9.1 it follows that any coset S-ring that contains A ,contains also A ′ . This proves the second equality and shows that A ≥ A ′ . Sinceobviously A ′ ≥ A , the first equality follows from equality (42) and Theorem 10.7. Proof of Theorem 1.2.
Without loss of generality we can assume that theS-ring A is schurian. Indeed, from Lemma 12.1 it follows that A = ( A ′ ) where A ′ = Sch( A ). Therefore it suffices to verify that(52) Φ ( A ) = Φ ( A ′ ) . However, by Theorem 9.1 any similarity ϕ ∈ Φ ( A ) is induced by an isomorphism f of A to itself. Without loss of generality we can assume that 1 f = 1. Then by (11)and the choice of ϕ we have X f = X ϕ = X for all X ∈ S ( A ). By Lemma 3.7this implies that f ∈ Aut( A ), and hence f ∈ Aut( A ′ ). Then ϕ is identical on A ′ ,and so ϕ ∈ Φ ( A ′ ). Thus Φ ( A ) ⊂ Φ ( A ′ ). Since the reverse inclusion is obvious,equality (52) follows.From the definition of the group Φ = Φ ( A ) it follows that A ≤ B where B = ( A ) Φ . To prove the reverse inclusion we have to verify that any basic set of A is contained in a basic set of B . Let x and y belong to the same basic set of A .Then it suffices to verify that(53) x δ = y for some δ ∈ Aut( B ) , δ = 1 . To do this we recall that A is schurian. So by Theorem 5.8 one can find a groupΓ ∈ M ( A ) such that(54) Γ S = Hol A ( S ) , S ∈ S ( A ) . It follows that there exists γ ∈ Γ such that 1 γ = 1 and x γ = y . Due to (54) wehave γ S ∈ Aut A ( S ) for all S ∈ S ( A ). By Lemma 11.2 this implies that the familyΣ = { γ S } is an S ( A )-multiplier of A . Therefore by Theorem 11.5 there exists auniquely determined similarity ϕ ∈ Φ such that Σ = Σ ϕ . This means that ϕ S isinduced by γ S for all S ∈ S ( A ). Since the S-ring A is separable, there exists anisomorphism γ of the ring A to itself that induces ϕ . Without loss of generalitywe can assume that 1 γ = 1. We claim that(55) x γ = y γ ′ for some γ ′ ∈ Aut( A ) , γ ′ = 1 . Then (53) holds for δ = γ ( γ ′ ) − because γ ∈ Aut( B ) by Lemma 3.7, and we aredone. Let us prove the claim. From Lemma 3.7 it follows that γ ∈ Aut( B ). Therefore x γ ∈ X where X is the basic set of A that contains x and y . Let S ∈ S ( A ).Then the bijections γ S and γ S induce the same similarity ϕ S of the S-ring ( A ) S .However, this S-ring equals Z S by statement (2) of Lemma 10.4. Therefore γ S = γ S .Thus the latter equality holds for all S ∈ S ( A ).Let now S be a section in the class \ S ( A ) that has h X i / rad( X ) as a subsection.Then S p ∈ S ( A ) for all primes p dividing | S | . By above this implies that x γ p L = ( x p L ) γ = ( x p L ) γ = x γp L = y p L where L is the denominator of S . On the other hand, the S-ring A S contains thetensor product of the S-rings A S p , and hence the permutation ( γ ) S equals theproduct of its p -projections. Thus x γ L = yL . By statement (2) of Theorem 10.11this implies that the elements x γ and y belong to the same basic set of A . Sincethis S-ring is schurian, there exists an automorphism γ ′ ∈ Aut( A ) such that x γ = y γ ′ and 1 γ ′ = 1. The claim is proved. Proof of Theorem 1.3.
To prove the “only if” part suppose that the S-ring A is schurian. Then by Theorem 5.8 there exists a group Γ ∈ M ( A ) that satisfies (54).Let S ∈ S . Then since S = \ S ( A ) (Corollary 10.10), we have S p ∈ S ( A ) forany prime p dividing | S | . Therefore due to (54) we conclude thatΓ S ≤ Y p Γ S p ≤ Y p Hol( S p ) = Hol( S ) . However, A S is the S-ring associated with the group Γ S because A is the S-ringassociated with the group Γ. Thus the S-ring A S is cyclotomic and condition (1)is satisfied. To verify condition (2) we note that Aut A ( S ) = Γ S . Therefore given σ ∈ Aut A ( S ) one can find γ ∈ Γ such that 1 γ = 1 and γ S = σ . Thus the element σ has a preimage Σ = { γ S } in the group Mult( A ).To prove the ”if part” suppose that for any section S ∈ S the S-ring A S is cy-clotomic and the restriction homomorphism from Mult( A ) to Aut A ( S ) is surjective.Then by the first assumption the class S is an A -admissible (Corollary 10.10) and(56) A S = Cyc(Aut A ( S ) , S ) = ( Z S ) Aut A ( S ) for all S ∈ S . Moreover, we claim that(57) ( Z S ) Aut A ( S ) = ( Z S ) (Φ ) S where Φ = Φ ( A ). Indeed, the left-hand side is obviously contained in the right-hand side. Suppose on the contrary that the reverse inclusion does not hold. Thenthere exists σ ∈ Aut A ( S ) such that ϕ σ (Φ ) S where ϕ σ is the similarity of Z S induced by σ . Moreover, by the surjectivity assumption there is a family Σ ∈ Mult( A ) the S -component of which equals σ . Since the class S is an A -admissible,Σ is an S -mulliplier of A . So by Theorem 11.5 there exists a similarity ϕ ∈ Φ such that ϕ S = ϕ σ . This implies that ϕ σ ∈ (Φ ) S . Contradition.Set A ′ = ( A ) Φ . Then from (56) and (57) it follows that(58) ( A ′ ) S = (( A ) Φ ) S = (( A ) S ) (Φ ) S = ( Z S ) (Φ ) S = A S for all S ∈ S . By Theorem 1.2 to complete the proof it suffices to verify that A = A ′ . To do this we note that A ≤ A ′ . So we only need to prove that A ≥ A ′ , i.e. that OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 39 any basic set X ′ ∈ S ( A ′ ) belongs to S ( A ). However, obviously S ′ = h X ′ i / rad( X ′ )is an A -section. Moreover, the extension ( A ) S ′ of ( A ′ ) S ′ has trivial radical byTheorem 5.7. By Lemma 5.3 this implies that S ′ ∈ S . Denote by X the basicset of the S-ring A h X ′ i that contains X ′ . Then since A S ′ = A ′ S ′ (see (58)), wehave π S ′ ( X ) = π S ′ ( X ′ ). It follows that X ⊂ X ′ rad( X ′ ) = X ′ . Thus X = X ′ asrequired. 13. Proof of Theorem 1.4
Reduction to quasidense S-rings.
Let A be an S-ring over a cyclic group G . Following paper [9] a class C of projectively equivalent A -sections is called singular if its rank is 2, its order is greater than 2 and it contains two sections S = L /L and T = U /U such that T is a multiple of S and the following twoconditions are satisfied:(S1) A is both the U /L - and U /L -wreath product,(S2) A U /L = A L /L ⊗ A U /L .By Lemma 6.2 of that paper S and T are necessarily the smallest and largest A -sections of C . Moreover, from Theorem 4.6 there, it follows that any rank 2 classof composite order belonging to P ( A ) is singular; in particular, the S-ring A isquasidense if and only if no class in P ( A ) is singular.For an A -section S we define the S -extension of A to be the smallest S-ring A ′ ≥ A such that A ′ S = Z S . From Theorem 4.2 it follows that A ′ does not dependon the choice of S in the class C ∈ P ( A ) of sections projectively equivalent to S .Denote this S-ring by E ( A , C ).The following lemma provides the reduction of Theorem 1.4 to the quasidensecase, and will be used later only for a singular class C of composite order. Lemma 13.1.
Let A be a circulant S-ring, C ∈ P ( A ) a singular class and A ′ = E ( A , C ) . Then (1) rk( A ′ ) > rk( A ) , (2) A ′ is schurian if and only if A is schurian. Proof . Statement (1) follows from the fact that any singular class has rank 2 andorder at least 3. To prove statement (2) denote by L /L and U /U the smallestand the largest A -sections of C (see Theorem 5.4). First, we claim that A ′ coincideswith any extension B of A that satisfies the following conditions(E1) B is both the U /L - and U /L -wreath product,(E2) B U = A U , B G \ L = A G \ L and B U /L = Z S ⊗ A U /L where S = L /L . Indeed, from condition (E2) it follows that B S = Z S . So by thedefinition of A ′ we have A ≤ A ′ ≤ B and ( A ′ ) S = Z S = B S . This implies that conditions (E1) and (E2) are satisfied with B replaced by A ′ . Itfollows that(59) B U = A ′ U , B G \ L = A ′ G \ L , B U /L = A ′ U /L , and A ′ and B are the U /L - and U /L -wreath products. Therefore to check that A ′ = B it suffices to verify that A ′ G/L = B G/L and A ′ U = B U . The former equality immediately follows from the second equality in (59). To verifythe latter one, we observe that A ′ U and B U are the U /L -wreath products. Thusthe required statement follows from the other two equalities in (59).To complete the proof we note that in terms of paper [9] every singular classis isolated (i.e. satisfies conditions (S1) and (S2)). Therefore by Lemma 6.5 ofthat paper the S-ring Ext C ( A , Z S ) defined there contains A and satisfies condi-tions (E1) and (E2). By the above claim this implies that Ext C ( A , Z S ) = A ′ .Thus statement (2) follows from [9, Theorem 6.7].We recall that b A is the S-ring dual to A , and b C is the class of projectivelyequivalent b A -sections that is dual to a class C ∈ P ( A ) (Subsection 2.3). Theorem 13.2.
Let A be a circulant S-ring and C a singular class of A . Then b C is a singular class of b A and the S-ring dual to E ( A , C ) coincides with E ( b A , b C ) . Proof . Let S ∈ C be a section of rank 2 and order at least 3. Then the class b C contains the section b S . Since | S | = | b S | and rk( b A b S ) = rk( A S ) = 2 (Subsection 2.3),the class b C has rank 2 and order greater than 2. Moreover, U ⊥ /U ⊥ and L ⊥ /L ⊥ are b A -sections, and statement (1) of Lemma 4.4 implies that the former one is amultiple of the latter. Finally, by Theorem 2.3 the S-ring b A satisfies conditions (S1)and (S2). Thus the class b C is singular.Let us prove that(60) E ( b A , b C ) ≤ \ E ( A , C ) . Indeed, since E ( A , C ) ≥ A , the S-ring dual to E ( A , C ) contains b A . Moreover,since E ( A , C ) S = Z S , the restriction of that ring to S ⊥ equals Z S ⊥ . Thus (60)follows from the definition of E ( b A , b C ). Next by duality, inclusion (60) also holdsafter interchanging A and b A . Therefore \ E ( A , C ) ≤ E ( b A , b C ) . Due to inclusion (60) this completes the proof of the theorem.Let us turn to the proof of Theorem 1.4. Let A be a circulant S-ring. First,we observe that given a singular class C ∈ P ( A ) of composite order the S-ring E ( A , C ) is schurian if and only if so is A (statement (2) of Lemma 13.1). Thereforeby statement (1) of that lemma and by Theorem 13.2 without loss of generalitywe can assume that the S-ring A has no singular classes of composite order, orequivalently that A is a quasidense S-ring (Theorem 5.5).13.2. Quasidense S-ring case.
Given σ ∈ Aut( G ) we set b σ to be the auto-morphism of b G taking a character χ to the character χ b σ : g χ ( g σ ). Then m ( σ ) = m ( b σ ) because χ b σ ( g ) = χ ( g σ ) = χ ( g m ) = χ ( g ) m , g ∈ G, where m = m ( σ ). Moreover, this shows that c σ S = b σ b S for any section S of G . Lemma 13.3.
Let A be a quasidense circular S-ring. Then (1) S ( b A ) = { b S : S ∈ S ( A ) } , (2) c A = b A and S ( b A ) = { b S : S ∈ S ( A ) } , OSET CLOSURE OF A CIRCULANT S-RING AND SCHURITY PROBLEM 41 (3) Aut b A ( b S ) = { b σ : σ ∈ Aut A ( S ) } for all S ∈ S ( A ) . Proof . Statement (1) follows from Theorem 5.10. From this statement andequivalence (1) ⇔ (4) of Theorem 8.3 it follows that a circulant S-ring is coset ifso is its dual. Therefore the set of duals to coset S-rings containing A coincideswith the set of coset S-rings containing b A . Thus c A = b A by the definition ofthe coset closure. The second part of statement (2) follows from the first one andstatement (1).To prove statement (3) let S ∈ S ( A ). Then the group Aut A ( S ) equals thelargest group K ≤ Aut( S ) for which Cyc( K, S ) ≥ A S . Since the dual to the S-ringCyc( K, S ) equals Cyc( b K, b S ) where b K = { b σ : σ ∈ K } (Theorem 2.2), the group b K is the largest subgroup of Aut( b S ) for which Cyc( b K, b S ) ≥ b A b S . So Aut b A ( b S ) = b K and we are done.To complete the proof of Theorem 1.4 let A be a schurian quasidence circulantS-ring. By duality we only have to prove that the S-ring b A is schurian. However, thelatter ring is quasidense by statement (2) of Theorem 5.9. Therefore by Theorem 1.3and statement (3) of Corollary 10.10 it suffices to verify that both of its conditionsare satisfied for the S-ring b A and sections belonging to S ( b A ).Let T ∈ S ( b A ). Then by statement (2) of Lemma 13.3 there exists a section S ∈ S ( A ) such that T = b S . Since A is schurian, the S-ring A S is cyclotomic(Theorem 1.3). Therefore the S-ring b A T = c A S is cyclotomic by Theorem 2.2. Thuscondition (1) is satisfied.To verify condition (2), let τ ∈ Aut b A ( T ). Since T ∈ S ( b A ) (Theorem 10.7),statement (3) of Lemma 13.3 implies that there exists σ ∈ Aut A ( S ) such that τ = b σ . Moreover, since A is schurian there exists an element Σ = { σ S ′ } of thegroup Mult( A ) such that σ S = σ (Theorem 1.3). Set b Σ = { σ T ′ } T ′ ∈ S ( b A ) where σ T ′ = c σ S ′ with S ′ defined by b S ′ = T ′ (see statement (2) of Lemma 13.3).Since c σ S = b σ = τ , it suffices to verify that b Σ ∈ Mult( b A ). By the remark afterDefinition 11.1 we have to verify that conditions (M1) and (M2) of that definitionare satisfied for the class S ( b A ) and family b Σ.Let τ ∈ Aut b A ( T ) where T ∈ S ( b A ). Then since Σ is a multiplier, for everysubsection T of T we have( τ ) T = ( c σ ) c S = \ ( σ ) S = c σ = τ where S i is such that T i = b S i and σ i ∈ Aut A ( S i ) is such that τ i = b σ i , i = 1 ,
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St. Petersburg Department of Steklov Institute of Mathematics, Russia
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