On the Table of Marks of a Direct Product of Finite Groups
aa r X i v : . [ m a t h . G R ] F e b ON THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITEGROUPS.
BRENDAN MASTERSON AND G ¨OTZ PFEIFFER
Abstract.
We present a method for computing the table of marks of a direct productof finite groups. In contrast to the character table of a direct product of two finite groups,its table of marks is not simply the Kronecker product of the tables of marks of the twogroups. Based on a decomposition of the inclusion order on the subgroup lattice of adirect product as a relation product of three smaller partial orders, we describe the tableof marks of the direct product essentially as a matrix product of three class incidencematrices. Each of these matrices is in turn described as a sparse block diagonal matrix.As an application, we use a variant of this matrix product to construct a ghost ringand a mark homomorphism for the rational double Burnside algebra of the symmetricgroup S . Introduction
The table of marks of a finite group G was first introduced by William Burnside in hisbook Theory of groups of finite order [5]. This table characterizes the actions of G ontransitive G -sets, which are in bijection to the conjugacy classes of subgroups of G . Thusthe table of marks provides a complete classification of the permutation representationsof a finite group G up to equivalence.The Burnside ring B ( G ) of G is the Grothendieck ring of the category of finite G -sets.The table of marks of G arises as the matrix of the mark homomorphism from B ( G ) tothe free Z -module Z r , where r is the number of conjugacy classes of subgroups of G . Likethe character table, the table of marks is an important invariant of the group G . By aclassical theorem of Dress [6], G is solvable if and only if the prime ideal spectrum of B ( G ) is connected, i.e., if B ( G ) has no nontrivial idempotents, a property that can easilybe derived from the table of marks of G .The table of marks of a finite group G can be determined by counting inclusions betweenconjugacy classes of subgroups of G [13]. For this, the subgroup lattice of G needs to beknown. As the cost of complete knowledge of the subgroups of G increases drasticallywith the order of G (or rather the number of prime factors of that order), this approachis limited to small groups. Alternative methods for the computation of a table of markshave been developed which avoid excessive computations with the subgroup lattice of G .This includes a method for computing the table of marks of G from the tables of marksof its maximal subgroups [13], and a method for computing the table of marks of a cyclicextension of G from the table of marks of G [12].The purpose of this article is to develop tools for the computation of the table of marksof a direct product of finite groups G and G . The obvious idea here is to relate thesubgroup lattice of G × G to the subgroup lattice of G and G , and to compute thetable of marks of G × G using this relationship. Many properties of G × G can bederived from the properties of G and G with little or no effort at all. Conjugacy classes Date : September 19, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Burnside ring, Table of marks, Subgroup lattice, Double Burnside ring, Ghostring, Mark homomorphism. of elements of G × G , for example, are simply pairs of conjugacy classes of G and G .And the character table of G × G is simply the Kronecker product of the charactertables of G and G . However the relationship between the table of marks of G × G and the tables of marks of G and G is much more intricate.A flavour of the complexity to be expected is already given by a classical result knownas Goursat’s Lemma (Lemma 2.1), according to which the subgroups of a direct productof finite groups G and G correspond to isomorphisms between sections of G and G .This article presents the first general and systematic study of the subgroup lattice of adirect product of finite groups beyond Goursat’s Lemma. Only very special cases of suchsubgroup lattices have been considered so far, e.g., by Schmidt [15] and Zacher [16].In view of Goursat’s Lemma, it seems appropriate to first develop some theory forsections in finite groups. Here, a section of a finite group G is a pair ( P, K ) of subgroups P, K of G such that K is a normal subgroup of P . We study sections by first defininga partial order on the set of sections of G as componentwise inclusion of subgroups: ( P ′ , K ′ ) ( P, K ) if P ′ P and K ′ K . Now, if ( P ′ , K ′ ) ( P, K ) , the canonical ho-momorphism P ′ /K ′ → P/K decomposes as a product of three maps: an epimorhism, anisomorphism and a monomorphism. We show that this induces a decomposition of thepartial order as a product of three partial orders, which we denote by K , P/K , and P for reasons that will become clear in Section 3. Thus = K ◦ P/K ◦ P , and this decomposition of the partial order is compatible with the conjugation action of G on the set of its sections.The description of subgroups of G × G in terms of sections of G and G allows usto transfer the decomposition of the partial orders on the sections to the set of subgroupsof G × G . We will show in Section 5 that, for subgroups L M of G × G , there existunique intermediary subgroups L ′ and M ′ such that L P L ′ P/K M ′ K M, where the partial orders P , P/K and K on the set of subgroups of G × G aredefined in terms of the corresponding relations on the sections of G and G . This givesa decomposition of the partial order on subgroups into three partial orders which iscompatible with the conjugation action of G × G . In Section 6, we will show as one ofour main results that this yields a corresponding decomposition of the table of marks of G as a matrix product of three class incidence matrices. Individually, each of these classincidence matrices has a block diagonal structure which is significantly easier to computethan the subgroup lattice of G × G .The rest of this paper is arranged as follows: In Section 2 we collect some useful knownresults. In Section 3 we study the sections of a finite group G and discuss properties ofthe lattice of sections, partially ordered componentwise. We show how a decompositionof this partial order as a relation product of three partial orders leads to a correspondingdecomposition of the class incidence matrix of the sections of G as a matrix product.This section concludes with a brief discussion of an interesting variant ′ of the partialorder on sections, and its class incidence matrix. Section 4 considers isomorphisms fromsections of G to a particular group U as subgroups of G × U . We determine the structureof the set of all such isomorphisms as a ( G, Aut ( U )) -biset. In Section 5, we study sub-groups of G × G as pairs of such isomorphisms, one from a section of G into U , andone from G . This allows us to determine the structure of the set of all such subgroupsas a ( G × G , Aut ( U )) -biset. We also derive a decomposition of the subgroup inclusionorder of G × G as a relation product of three partial orders from the corresponding N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 3 decomposition of the partial orders of sections from Section 3. In Section 6 we developmethods for computing the individual class incidence matrices for each of the partialorders on subgroups and use these matrices to compute the table of marks of G × G ,essentially as their product. Finally, in Section 7 we present an application of the theory.The double Burnside ring B ( G, G ) of a finite group G is defined as the Grothendieck ringof transitive ( G, G ) -bisets and, where addition is defined as disjoint union and multipli-cation is tensor product. The double Burnside ring is currently at the centre of muchresearch and is an important invariant of the group G , see e.g. [1, 2, 4, 14]. Here westudy the particular case of G = S , and use our partial orders to construct an explicitghost ring and mark homomorphism for Q B ( G, G ) , in the sense of Boltje and Danz [1]. Acknowledgement:
Much of the work in this article is based on the first author’sPhD thesis (see [11]). This research was supported by the College of Science, NationalUniversity of Ireland, Galway. 2.
Preliminaries
Notation.
We denote the symmetric group of degree n by S n , the alternating groupof degree n by A n , and a cyclic group of order n simply by n .We use various forms of composition in this paper. Group homomorphisms act fromthe right and are composed accordingly: the product of φ : G → G and ψ : G → G is φ · ψ : G → G , defined by a φ · ψ = ( a φ ) ψ , for a ∈ G , where G i is a group, i =
1, 2, 3 .The relation product of relations R ⊆ X × Y and S ⊆ Y × Z is the relation S ◦ R = { ( x, z ) : ( x, y ) ∈ R and ( y, z ) ∈ S for some y ∈ Y } ⊆ X × Z , where X, Y, Z are sets.In section 2.2, the product L ∗ M of subgroups L G × G and M G × G will bedefined as ( M op ◦ L op ) op , where R op = { ( y, x ) : ( x, y ) ∈ R } denotes the opposite of R .2.2. Subgroups as Relations.
The following classical result describes subgroups of adirect product as isomorphisms between section quotients. Here, a section of a finitegroup G is a pair ( P, K ) of subgroups of G so that K E P . Lemma 2.1 (Goursat’s Lemma, [7]) . Let G , G be groups. There is a bijective corre-spondence between the subgroups of the direct product G × G and the isomorphisms ofthe form θ : P /K → P /K , where ( P i , K i ) is a section of G i , i =
1, 2 .Proof.
Let L G × G and let P i G i be the projection of L onto G i , i =
1, 2 . Then L is a binary relation from P to P . Writing a La for ( a , a ) ∈ L , it is easy to seethat {{ a ∈ P : a La } : a ∈ P } is a partition of P into cosets of the normal subgroup K = { a ∈ G : } of P . Similarly, the sets { a ∈ P : a La } , a ∈ P , are cosetsof a normal subgroup K of P . The relation L thus is difunctional, i.e., it establishesa bijection θ between the section quotients P /K and P /K , which in fact is a grouphomomorphism.Conversely, any isomorphism θ : P /K → P /K between sections ( P i , K i ) of G i , i =
1, 2 , yields a relation { ( a , a ) ∈ G × G : ( a K ) θ = a K } , which in fact is a subgroupof G × G . (cid:3) If a subgroup L corresponds to an isomorphism θ : P /K → P /K , then we write p i ( L ) for P i and k i ( L ) for K i , i =
1, 2 . We call the sections ( P i , K i ) the Goursat sections of L and the isomorphism type of P i /K i the Goursat type of L . Finally, L is called the graph of θ and, conversely, θ is the Goursat isomorphism of L .The next lemma, illustrated in Fig. 1, can be derived from Lemma 2.1, see e.g. [9]. BRENDAN MASTERSON AND G ¨OTZ PFEIFFER ( K ∩ P )( P ∩ K ) P ∩ P K ∩ P K ∩ P K K K ′ K ′ P ′ P ′ P P Figure 1.
Butterfly Lemma
Lemma 2.2 (Butterfly Lemma, [8, 11.3]) . Let ( P , K ) and ( P , K ) be sections of G .Set P ′ i := ( P ∩ P ) K i for i =
1, 2 , K ′ := ( P ∩ K ) K , and K ′ := ( P ∩ K ) K . Then P ∩ P = P ′ ∩ P ′ , ( K ∩ P )( P ∩ K ) = K ′ ∩ K ′ and the canonical map φ i : ( P ′ ∩ P ′ ) / ( K ′ ∩ K ′ ) → P ′ i /K ′ i is an isomorphism, i =
1, 2 . We refer to the section ( P ′ ∩ P ′ , K ′ ∩ K ′ ) as the Butterfly meet of ( P , K ) and ( P , K ) .Let G , G , G be finite groups. The product L ∗ M of subgroups L G × G and M G × G is defined as L ∗ M = { ( g , g ) ∈ G × G : ( g , g ) ∈ L and ( g , g ) ∈ M for some g ∈ G } .Then L ∗ M ⊆ G × G is in fact a subgroup. Thanks to [2], we obtain the Goursatisomorphism of L ∗ M by composing Goursat isomorphisms θ ′ and ψ ′ , as follows. Supposethat L is the graph of the isomorphism θ : P /K → P /K , and that M is the graph of ψ : P /K → P /K . With both ( P , K ) and ( P , K ) being sections of G , let subgroups P ′ i , K ′ i , and isomorphisms φ i : ( P ′ ∩ P ′ ) / ( K ′ ∩ K ′ ) → P ′ i /K ′ i , i =
1, 2 , be as in the ButterflyLemma 2.2. Let ψ : P ′ /K ′ → P ′ /K ′ be the isomorphism obtained by restricting ψ to P ′ /K ′ , defined by ( pK ′ ) ψ = ( pK ′ ) ψ for p ∈ P ′ . Moreover, let θ : P ′ /K ′ → P ′ /K ′ be theco-restriction of θ to P ′ /K ′ , defined by ( pK ′ ) θ = ( pK ′ ) θ for p ∈ P ′ . Then the graph of θ ′ := θ · φ − : P ′ /K ′ → ( P ′ ∩ P ′ ) / ( K ′ ∩ K ′ ) is a subgroup of G × G (although not necessarily of L ), the graph of ψ ′ := φ · ψ : ( P ′ ∩ P ′ ) / ( K ′ ∩ K ′ ) → P ′ /K ′ is a subgroup of G × G . Lemma 2.3.
With the above notation, L ∗ M is the graph of the composite isomorphism θ ′ · ψ ′ : P ′ /K ′ → P ′ /K ′ . We use the subgroup product and its Goursat isomorphism in the proof of Theorem 6.5.2.3.
Bisets and Biset Products.
The action of a direct product G × G on a set X issometimes more conveniently described as the two groups G i acting on the same set X ,one from the left and one from the right. N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 5
Definition 2.4 ([3, 2.3.1]) . Let G and G be groups. Then a ( G , G ) - biset X is a left G -set and a right G -set, such that the actions commute, i.e., ( g x ) g = g ( xg ) , g i ∈ G i , x ∈ X .Under suitable conditions, bisets can be composed, as follows. Definition 2.5.
Let G , G and G be groups. If X is a ( G , G ) -biset and Y a ( G , G ) -biset, the tensor product of X and Y is the ( G , G ) -biset X × G Y := ( X × Y ) /G of G -orbits on the set X × Y under the action given by ( x, y ) .g = ( x.g, g − .y ) , g ∈ G .The tensor product of bisets will be used in Section 5 to describe certain sets of sub-groups of G × G . It also provides the multiplication in the double Burnside ring of agroup G , which is the subject of Section 7.1.2.4. Action on Pairs.
We will also need to deal with one group acting on two sets. Thefollowing parametrization of the orbits of a group acting on a set of pairs is well-known.
Lemma 2.6.
Let G be a finite group, acting on finite sets X and Y , and suppose that Z ⊆ X × Y is a G -invariant set of pairs. Then Z/G = a [ y ] G ∈ Y/G { [ x, y ] G : [ x ] G y ∈ Zy/G y } , where Zy = { x ∈ X : ( x, y ) ∈ Z } for y ∈ Y . The G -orbits of pairs in Z are thus represented by pairs ( x, y ) , where the y representthe orbits of G on Y and, for a fixed y , the x represent the orbits of the stabilizer of y on the set Zy of all x ∈ X that are Z -related to y . Proof.
Note that Z = a [ y ] G ∈ Y/G Z ∩ ( X × [ y ] G ) is a disjoint union of G -invariant intersections Z ∩ ( X × [ y ] G ) , whence Z/G is the corre-sponding disjoint union of orbit spaces ( Z ∩ ( X × [ y ] G )) /G . By [12, Lemma 2.1] , for each y ∈ Y , the map [ x ] G y [ x, y ] G is a bijection between X/G y and ( X × [ y ] G ) /G . Hence, for every y ∈ Y , there is a bijectionbetween Zy/G y and ( Z ∩ ( X × [ y ] G )) /G , (cid:3) Class Incidence Matrices.
Let ( X, ) be a finite partially ordered set (poset)with incidence matrix A ( ) = ( a xy ) x,y ∈ X , where a xy = (cid:14) , if y x , , else.This incidence matrix A ( ) is lower triangular, if the order of rows and columns of A ( ) extends the partial order .Suppose further that ≡ is an equivalence relation on X . Then ≡ partitions X intoclasses X/ ≡ = { [ x ] : x ∈ T } , for a transversal T ⊆ X . We say that the partial order is compatible with the equivalence relation ≡ if, for all classes [ x ] , [ y ] , the number a xy := { x ′ ≡ x : y x ′ } BRENDAN MASTERSON AND G ¨OTZ PFEIFFER does not depend on the choice of the representatives x, y ∈ X , i.e., if a xy = a xy ′ for y ′ ≡ y . In that case, we define the class incidence matrix of the partial order to bethe matrix A ( ) = ( a xy ) x,y ∈ T ,whose rows and columns are labelled by the chosen transversal T . Matrix multiplicationrelates the matrices A ( ) and A ( ) in the following way. Lemma 2.7.
Define a row summing matrix R ( ≡ ) = ( r xy ) x ∈ T, y ∈ X and a column pickingmatrix C ( ≡ ) = ( c xy ) x ∈ X,y ∈ T with entries r xy = (cid:14) , if x ≡ y , , else, c xy = (cid:14) , if x = y , , else.Then (i) R ( ≡ ) · C ( ≡ ) = I , the identity matrix on T . (ii) R ( ≡ ) · A ( ) = A ( ) · R ( ≡ ) . (iii) A ( ) = R ( ≡ ) · A ( ) · C ( ≡ ) ,Proof. (i) For each x, z ∈ T , P y ∈ X r xy c yz = r xz . (ii) For each x ∈ T , z ∈ X , the x, z -entryof both matrices is equal to a xy , where y ∈ T represents the class z ∈ X . (iii) followsfrom (ii) and (i). (cid:3) Remark 2.8.
Examples of compatible posets are provided by group actions. Supposethat a finite group G acts on a poset ( X, ) in such a way that x y = ⇒ x.a y.a for all x, y ∈ X and all a ∈ G . Then X is called a G -poset . The partial order iscompatible with the partition of X into G -orbits since { x ′ ≡ x : y x ′ } .a = { x ′ ≡ x : y.a x ′ } ,for all x, y ∈ X . We write R ( G ) and C ( G ) for R ( ≡ ) and C ( ≡ ) if the equivalence ≡ isgiven by a G -action. Remark 2.9.
More generally, any square matrix A with rows and columns indexed bya set X with an equivalence relation ≡ , after choosing a transversal of the equivalenceclasses, yields a product R ( ≡ ) · A · C ( ≡ ) . We say that the matrix A is compatible withthe equivalence if this product does not depend on the choice of transversal.If the equivalence on X is induced by the action of a group G then the matrix A =( a xy ) x,y ∈ X is compatible if a x.g,y.g = a xy for all g ∈ G . Such matrices are the subjectof Proposition 4.7 and Theorem 6.5.2.6. The Burnside Ring and the Table of Marks.
The
Burnside ring B ( G ) of a finitegroup G is the Grothendieck ring of the category of finite G -sets, that is the free abeliangroup with basis consisting of the isomorphism classes [ X ] of transitive G -sets X , withdisjoint union as addition and the Cartesian product as multiplication. Multiplication oftransitive G -sets is described by Mackey’s formula [10, Lemma 1.2.11] [ G/A ] · [ G/B ] = X ` d AdB = G [ G/ ( A d ∩ B )] .The rational Burnside algebra Q B ( G ) = Q ⊗ Z B ( G ) is isomorphic to a direct sum of r copies of Q , one for each conjugacy class of subgroups of G , with products of basiselements determined by the above formula. N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 7
The mark of a subgroup H of G on a G -set X is its number of fixed points, | X H | = { x ∈ X : x.h = x for all h ∈ H } . Obviously, | X H | = | X H | whenever H and H areconjugate subgroups of G . The map β G : B ( G ) → Z r assigns to [ X ] ∈ B ( G ) the vector ( | X H | , . . . , | X H r | ) ∈ Z r , where ( H , . . . , H r ) is a transversal of the conjugacy classes ofsubgroups of G . In this context, the ring Z r with componentwise addition and multipli-cation, is called the ghost ring of G . We have β G ([ X ∐ Y ]) = β G ([ X ]) + β G ([ Y ]) , β G ([ X × Y ]) = β G ([ X ]) · β G ([ Y ]) , where the latter product is componentwise multiplication in Z r . Thus β G is a homomor-phism of rings, called the mark homomorphism of G .The table of marks M ( G ) of G is the r × r -matrix with rows β G ([ G/H i ]) , i =
1, . . . , r ,the mark vectors of all transitive G -sets, up to isomorphism. Regarding β G as a linearmap from Q B ( G ) to Q r , the table of marks is the matrix of β G relative to the naturalbasis ([ G/H i ]) of Q B ( G ) and the standard basis of Q r .As | ( G/H ) K | = | N G ( H ) : H | { H g > K : g ∈ G } for subgroups H, K G , the table ofmarks provides a compact description of the subgroup lattice of G . In fact M ( G ) = D · A ( ) ,where D is the diagonal matrix with entries | N G ( H i ) : H i | and A ( ) is the class incidencematrix of the group G acting on its lattice of subgroups by conjugation. Example 2.10.
Let G = S . Then G has conjugacy classes of subgroups and M ( G ) = G/1 6 . . .G/2 3 1 . .G/3 2 . 2 .G/G 1 1 1 1 = · . Sections
Let G be a finite group. We denote by S G the set of subgroups of G , and by S G /G := { [ H ] G : H G } the set of conjugacy classes of subgroups of G . A section of G is a pair ( P, K ) of subgroupsof G where K E P . We call P the top group and K the bottom group of the section ( P, K ) .We refer to the quotient group P/K as the quotient of the section ( P, K ) . The isomorphismtype of a section is the isomorphism type of its quotient and the size of a section is thesize its quotient. We denote the set of sections of G by Q G := { ( P, K ) : K E P G } .The group G acts on the set of pairs Q G by conjugation. In Sections 3.1 and 3.2, we classifythe orbits of this action and describe the automorphisms induced by the stabilizer of asection on its quotient. The partial order on S G induces a partial order on the pairs in Q G . In Section 3.3, we show that this partial order is in fact a lattice, and how it canbe decomposed as a product of three smaller partial order relations. In Section 3.4,wedetermine the class incidence matrix of Q G and show that the decomposition of thepartial order on Q G corresponds to a decomposition of the class incidence matrix of Q G as a matrix product of three class incidence matrices. In Section 3.5, we use the smallerpartial orders to define a new partial order on Q G that is consistent with the notion ofsize of a section. BRENDAN MASTERSON AND G ¨OTZ PFEIFFER
Conjugacy Classes of Sections.
A finite group G naturally acts on its sectionsthrough componentwise conjugation via ( P, K ) g := ( P g , K g ) ,where ( P, K ) ∈ Q G and g ∈ G . We write [ P, K ] G for the conjugacy class of a section ( P, K ) in G , and denote the set of all conjugacy classes of sections of G by Q G /G := { [ P, K ] G : ( P, K ) ∈ Q G } .The conjugacy classes of sections can be parametrized in different ways in terms of simpleractions, as follows. Proposition 3.1.
Let G and S G be as above. (i) For P G , let S E PG = { K ∈ S G : K E P } . Then ( S E PG , ) is an N G ( P ) -poset and Q G /G = a [ P ] ∈ S G /G { [ P, K ] G : [ K ] ∈ S E PG /N G ( P ) } . (ii) For K G , let S K E G = { P ∈ S G : K E P } . Then ( S K E G , ) is an N G ( K ) -poset and Q G /G = a [ K ] ∈ S G /G { [ P, K ] G : [ K ] ∈ S K E G /N G ( K ) } . Proof. (i) Note that Q G ⊆ S G × S G is a G -invariant set of pairs. As the stabilizer of K ∈ S G is N G ( K ) , the result follows with Lemma 2.6. (ii) Follows in a similar way. (cid:3) We write U ⊑ G for a finite group U which is isomorphic to a subquotient of G . Wedenote by Q G ( U ) the set of sections of G with isomorphism type U , and by Q G ( U ) /G := { [ P, K ] G ∈ Q G /G : P/K ∼ = U } . its G -conjugacy classes. Naturally, Q G /G = a U ⊑ G Q G ( U ) /G .Each of the above three partitions of Q G /G will be used in the sequel.3.2. Section Automizers.
The automizer of a subgroup H in G is the quotient groupof the section ( N G ( H ) , C G ( H )) . The automizer of H is isomorphic to the subgroup ofAut ( H ) induced by the conjugation action of G . Analogously, we define the automizerof a section as a section whose quotient is isomorphic to the subgroup of automorphismsinduced by conjugation by G . Definition 3.2.
Let ( P, K ) ∈ Q G and set N = N G ( K ) . Using the natural homomorphism φ : N → N/K, n n = nK ,we let P := φ ( P ) = P/K and N := φ ( N ) = N/K . We define the section normalizer of ( P, K ) to be the inverse image N G ( P, K ) := φ − ( N N ( P )) , the section centralizer to be C G ( P, K ) := φ − ( C N ( P )) , and the section automizer to be the section A G ( P, K ) := ( N G ( P, K ) , C G ( P, K )) . Moreover, we denote by Aut G ( P, K ) the subgroup of Aut ( P/K ) of automorphisms inducedby conjugation by G , see Fig. 2. N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 9
KPP ∩ C G ( P, K ) C G ( P, K ) N G ( P, K ) P C G ( P, K ) ∩ C N ( P ) C N ( P ) N N ( P ) P C N ( P ) Inn ( P/K ) Aut G ( P, K ) Aut ( P/K ) Figure 2.
The section ( P, K ) and its automorphismsThe following properties of these groups are obvious. Lemma 3.3.
Let ( P, K ) be a section in Q G . Then (i) N G ( P, K ) = N G ( P ) ∩ N G ( K ) . (ii) C G ( P, K ) is the set of all g ∈ N G ( P, K ) which induce the identity automorphismon P/K . (iii) Inn ( G ) Aut G ( P, K ) Aut ( P/K ) . The Sections Lattice.
Subgroup inclusion induces a partial order on the set Q G of sections of G which inherits the lattice property from the subgroup lattice, as follows. Definition 3.4. Q G is a poset, with partial order defined componentwise, i.e., ( P ′ , K ′ ) ( P, K ) if P ′ P and K ′ K ,for sections ( P ′ , K ′ ) and ( P, K ) of G .For subgroups A, B G , we write A ∨ B = h A, B i for the join of A and B in thesubgroup lattice of G , and hh A ii B for the normal closure of A in B . Proposition 3.5.
The poset ( Q G , ) is a lattice with componentwise meet, i.e., ( P , K ) ∩ ( P , K ) = ( P ∩ P , K ∩ K ) ,and join given by ( P , K ) ∨ ( P , K ) = ( P ∨ P , hh K ∨ K ii P ∨ P ) ,for sections ( P , K ) and ( P , K ) of G .Proof. Clearly, K ∩ K is a normal subgroup of P ∩ P , and the section ( P ∩ P , K ∩ K ) is the unique greatest lower bound of the sections ( P , K ) and ( P , K ) in Q G .It is also easy to see that the least section ( P, K ) of G with P > P ∨ P and K > K ∨ K has P = P ∨ P and K = hh K ∨ K ii P . (cid:3) Theorem 3.6.
Let ( P ′ , K ′ ) ( P, K ) be sections of a finite group G . Then (i) ( P ′ , K ∩ P ′ ) is the largest section between ( P ′ , K ′ ) and ( P, K ) with top group P ′ ; (ii) ( P ′ K, K ) is the smallest section between ( P ′ , K ′ ) and ( P, K ) with bottom group K ; (iii) the map p ( K ∩ P ′ ) pK , p ∈ P ′ is an isomorphism between the section quotientsof ( P ′ , K ∩ P ′ ) and ( P ′ K, K ) . PP ′ KKP ′ K ∩ P ′ K ′ Figure 3. ( P ′ , K ′ ) ( P, K ) Proof. If ( P ′ , K ′ ) ( P, K ) then there is a canonical homomorphism from P ′ /K ′ to P/K ,given by ( K ′ p ) φ = Kp for p ∈ P ′ . According to the homomorphism theorem, φ can bedecomposed into a surjective, bijective and an injective part, that is φ = φ φ φ , where φ : P ′ /K ′ → P ′ /K ∩ P ′ , φ : P ′ /K ∩ P ′ → P ′ K/K, φ : P ′ K/K → P/K are uniquely determined, see Fig. 3 (cid:3)
Motivated by the above result we define the following three partial orders on Q G . Definition 3.7.
Let ( P ′ , K ′ ) ( P, K ) . Then we write(i) ( P ′ , K ′ ) P ( P, K ) if P ′ = P , i.e., if the sections have the same top groups;(ii) ( P ′ , K ′ ) K ( P, K ) if K ′ = K , i.e., if the sections have the same bottom groups;(iii) ( P ′ , K ′ ) P/K ( P, K ) if the map pK ′ pK , p ∈ P , is an isomorphism.We can now reformulate Theorem 3.6 in terms of these three relations. Corollary 3.8.
The partial order on Q G is a product of three relations, i.e., = K ◦ P/K ◦ P .Let A ( ) denote the incidence matrix for the partial order . Then the stronger property A ( ) = A ( K ) · A ( P/K ) · A ( P ) also holds.Proof. By Theorem 3.6, for ( P ′ , K ′ ) ( P, K ) there exists unique intermediate sections S, S ′ ∈ Q G such that ( P ′ , K ′ ) P S ′ P/K S K ( P, K ) . (cid:3) Remark 3.9.
Note that, by the Correspondence Theorem, there is a bijective correspon-dence between the subgroups of
P/K and the sections ( P ′ , K ′ ) of G with ( P ′ , K ′ ) K ( P, K ) .Similarly, there is a bijective correspondence between the normal subgroups (and hencethe factor groups) of P/K and the sections ( P ′ , K ′ ) of G with ( P, K ) P ( P ′ , K ′ ) .3.4. Class Incidence Matrices.
We denote the class incidence matrix of the G -poset ( Q G , ) by A ( ) . Note that the set Q G of sections of G is also a G -poset with respectto any of the partial orders from Definition 3.7, with respective class incidence matrices A ( P ) , A ( K ) and A ( P/K ) . Theorem 3.10.
With this notation, A ( ) = A ( K ) · A ( P/K ) · A ( P ) . N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 11
Proof.
Set R = R ( G ) and C = C ( G ) . From Lemma 2.7(iii) we have that A ( ) = R · A ( ) · C . By Corollary 3.8 A ( ) = A ( K ) · A ( P/K ) · A ( P ) . Lemma 2.7(ii) then gives A ( ) = R · A ( K ) · A ( P/K ) · A ( P ) · C = A ( K ) · R · A ( P/K ) · A ( P ) · C = A ( K ) · A ( P/K ) · R · A ( P ) · C = A ( K ) · A ( P/K ) · A ( P ) . (cid:3) Each of the classes incidence matrices A ( P ) , A ( K ) and A ( P/K ) is a direct sum ofsmaller class incidence matrices, as the following results show. Theorem 3.11.
For P G , denote the class incidence matrix of the N G ( P ) -poset S E PG by A P ( ) . Then A ( P ) = M [ P ] ∈ S G /G A P ( ) .Proof. Let ( P, K ) ∈ Q G . By Proposition 3.1, the G -conjugacy classes containing a sectionwith top group P are represented by sections ( P, K ′ ) , where K ′ runs over a transversal ofthe N G ( P ) -orbits of S E PG . In order to count the G -conjugates of ( P, K ′ ) above ( P, K ) inthe P -order, it now suffices to note that ( P, K ) ( P, K ′ ) g for some g ∈ G if and only if K ( K ′ ) g for some g ∈ N G ( P ) . (cid:3) Example 3.12.
Let G = S . Then A ( P ) = (
1, 1 ) · · · · · · · (
2, 1 ) · · · · · · · (
2, 2 ) · · · · · · (
3, 1 ) · · · · · · · (
3, 3 ) · · · · · · ( G, 1 ) · · · · · · · ( G, 3 ) · · · · · · ( G, G ) · · · · · (
1, 1 ) (
2, 1 ) (
2, 2 ) (
3, 1 ) (
3, 3 ) (
G, 1 ) (
G, 3 ) (
G, G ) Theorem 3.13.
For K G , denote the class incidence matrix of the N G ( K ) -poset S K E G by A K ( ) . Then A ( K ) = M [ K ] ∈ S G /G A K ( ) .Proof. Similar to the proof of Theorem 3.11. (cid:3)
Example 3.14.
Let G = S . Then A ( K ) = (
1, 1 ) · · · · · · · (
2, 1 ) · · · · · · (
3, 1 ) · · · · · · ( G, 1 ) · · · · (
2, 2 ) · · · · · · · (
3, 3 ) · · · · · · · ( G, 3 ) · · · · · · ( G, G ) · · · · · · · (
1, 1 ) (
2, 1 ) (
3, 1 ) (
G, 1 ) (
2, 2 ) (
3, 3 ) (
G, 3 ) (
G, G ) Lemma 3.15.
We have A ( P/K ) = M U ⊑ G A U ( P/K ) where, for U ⊑ G , A U ( P/K ) is the class incidence matrix of the G -poset ( Q G ( U ) , P/K ) . Proof. ( P ′ , K ′ ) P/K ( P, K ) implies P ′ /K ′ ∼ = P/K . (cid:3) Example 3.16.
Let G = S . Then A ( P/K ) = (
1, 1 ) · · · · · · · (
2, 2 ) · · · · · · (
3, 3 ) · · · · · · ( G, G ) · · · · (
2, 1 ) · · · · · · · ( G, 3 ) · · · · · · (
3, 1 ) · · · · · · · ( G, 1 ) · · · · · · · (
1, 1 ) (
2, 2 ) (
3, 3 ) (
G, G ) (
2, 1 ) (
G, 3 ) (
3, 1 ) (
G, 1 ) The class incidence matrix A ( ) of the G -poset ( Q G , ) is the product of this matrixand the class incidence matrices in Examples 3.14 and 3.12, according to Theorem 3.10: A ( ) = (
1, 1 ) · · · · · · · (
2, 2 ) · · · · · (
3, 3 ) · · · · · ( G, G ) (
2, 1 ) · · · · · · ( G, 3 ) · · (
3, 1 ) · · · · · · ( G, 1 ) · · · · (
1, 1 ) (
2, 2 ) (
3, 3 ) (
G, G ) (
2, 1 ) (
G, 3 ) (
3, 1 ) (
G, 1 ) The Sections Lattice Revisited.
The partial order = K ◦ P/K ◦ P on Q G is not compatible with section size as ( P ′ , K ′ ) P ( P, K ) implies | P ′ /K ′ | > | P/K | . It turnsout that, by effectively replacing the partial order P by its opposite > P , one obtainsfrom a new partial order ′ , which is compatible with section size. Proposition 3.17.
Define a relation ′ on Q G by ( P ′ , K ′ ) ′ ( P, K ) if P ′ P and K ∩ P ′ K ′ for sections ( P ′ , K ′ ) and ( P, K ) of G . Then ( Q G , ′ ) is a G -poset.Proof. The relation ′ is clearly reflexive and antisymmetric on Q G , and compatible withthe action of G . Hence it only remains to be shown that this relation is transitive.Let ( P ′′ , K ′′ ) , ( P ′ , K ′ ) and ( P, K ) be sections of G , such that ( P ′′ , K ′′ ) ′ ( P ′ , K ′ ) and ( P ′ , K ′ ) ′ ( P, K ) . In order to show that ( P ′′ , K ′′ ) ′ ( P, K ) , we need P ′′ P (whichis clear), and K ∩ P ′′ K ′′ . Intersecting both sides of K ∩ P ′ K ′ with P ′′ gives K ∩ P ′′ K ′ ∩ P ′′ K ′′ , as desired. (cid:3) Example 3.18.
Let us denote the three subgroups of order of the Klein -group G = by , and . Then ( , 1 ) ′ ( G, 2 ) , ( G, 2 ) and ( G, G ) ′ ( G, 2 ) , ( G, 2 ) . As thesections ( G, 2 ) , ( G, 2 ) have no unique infimum the poset ( Q G , ′ ) is not a lattice. Proposition 3.19.
Let ( P ′ , K ′ ) and ( P, K ) be sections of a finite group G , such that ( P ′ , K ′ ) ′ ( P, K ) . Then, there are uniquely determined sections of G , ( P, K ) > ′ ( P , K ) > ′ ( P , K ) > ′ ( P ′ , K ′ ) such that (i) ( P , K ) K ( P, K ) , (ii) ( P ′ , K ′ ) > P ( P , K ) , (iii) ( P , K ) P/K ( P , K ) .Proof. By definition, ( P ′ , K ′ ) ′ ( P, K ) implies P ′ P and K ∩ P ′ K ′ , where K ′ E P ′ and K E P . Then, by the second isomorphism theorem, K ∩ P ′ is a normal subgroup of P ′ , P ′ K is a subgroup of P such that K E P ′ K and ( P ′ K ) /K is isomorphic to P ′ / ( K ∩ P ′ ) . N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 13 PP ′ KK ′ KKP ′ K ′ K ∩ P ′ Figure 4. ( P ′ , K ′ ) ′ ( P, K ) Hence ( P , K ) = ( P ′ K, K ) and ( P , K ) = ( P ′ , K ∩ P ′ ) have the desired properties, seeFig. 4. (cid:3) Corollary 3.20.
The partial order ′ on Q G is a product of three relations, i.e., ′ = K ◦ P/K ◦ > P .Moreover, A ( ′ ) = A ( K ) · A ( P/K ) · A ( > P ) and A ( ′ ) = A ( K ) · A ( P/K ) · A ( > P ) . Example 3.21.
Let G = S . Then A ( ′ ) = (
1, 1 ) · · · · · · · (
2, 2 ) · · · · · · (
3, 3 ) · · · · · · ( G, G ) · · · · (
2, 1 ) · · · · · ( G, 3 ) · · (
3, 1 ) · · · · · ( G, 1 ) (
1, 1 ) (
2, 2 ) (
3, 3 ) (
G, G ) (
2, 1 ) (
G, 3 ) (
3, 1 ) (
G, 1 ) In contrast to the class incidence matrix A ( ) in Example 3.16, the matrix A ( ′ ) islower triangular when rows and columns are sorted by section size. Moreover, ( P, K ) ′ ( G, 1 ) , for all sections ( P, K ) of G . Remark 3.22.
Whenever ( P ′ , K ′ ) ′ ( P, K ) , there is a canonical isomorphism ψ : P ′ /K ′ → P ′ K/K ′ K . Let θ : P /K → P /K be the Goursat isomorphism of a subgroup of G × G and suppose that ( P ′ , K ′ ) ′ ( P , K ) . The canonical isomorphism determines a unique restriction of θ to a Goursat isomorphism θ ′ : P ′ /K ′ → P ′ /K ′ . Similarly, for each sec-tion ( P ′ , K ′ ) ′ ( P , K ) , there is a unique co-restriction of θ to a Goursat isomorphism θ ′ : P ′ /K ′ → P ′ /K ′ . As the Butterfly meet ( P ′ , K ′ ) of sections ( P , K ) and ( P , K ) of agroup G satisfies ( P ′ , K ′ ) ′ ( P , K i ) , i =
1, 2 , by Lemma 2.3, the product of subgroupswith Goursat isomorphisms θ : P /K → P /K and ψ : P /K → P /K is the composi-tion of the restriction of θ and the co-restriction of ψ to the Butterfly meet of ( P , K ) and ( P , K ) . 4. Morphisms
Let U be a finite group. A U -morphism of G is an isomorphism θ : P/K → U betweena section ( P, K ) of G and the group U . The set M G ( U ) := { θ : P/K → U | ( P, K ) ∈ Q G ( U ) } of all U -morphisms of G forms a ( G, Aut ( U )) -biset. In Section 4.1, we describe the set M G ( U ) /G of G -classes of U -morphisms as an Out ( U ) -set. The identification of M G ( U ) with certain subgroups of G × U in Section 4.2 induces a partial order on M G ( U ) . InSection 4.3, we compute the class incidence matrix of this partial order.4.1. Classes of U -Morphisms. Each U -morphism θ : P/K → U of G induces an iso-morphism between the automorphism groups Aut ( P/K ) and Aut ( U ) . We define theautomizer of θ as an isomorphism between the quotient of the automizer A G ( P, K ) of thesection ( P, K ) and the corresponding subgroup of Aut ( U ) . Definition 4.1.
Given a U -morphism θ : ( P, K ) → U , denote ˜ P = N G ( P, K ) and ˜ K = C G ( P, K ) and let A θ Aut ( U ) be the image of Aut G ( P/K ) in Aut ( U ) . The automizer of the U -morphism θ is the A θ -morphism A G ( θ ) : ˜ P/ ˜ K → A θ ,that, for n ∈ ˜ P , maps the coset n ˜ K to the automorphism θ − γ n θ of U corresponding toconjugation by n on P/K . Moreover, denote by O θ := A θ / Inn ( U ) Out ( U ) ,the group of outer automorphisms of U induced via θ , noting that Inn ( U ) A θ .The group G acts on M G ( U ) via θ a = γ − θ , where γ a : P/K → P a /K a is the conjuga-tion map induced by a ∈ G . We denote by [ θ ] G := { θ a : a ∈ G } the G -orbit of the U -morphism θ and by M G ( U ) /G := { [ θ ] G : θ ∈ M G ( U ) } the set of G -classes of U -morphisms.For a section ( P, K ) ∈ Q G ( U ) , denote by M P,KG ( U ) the set of U -morphisms with domain P/K . Under the action ( θ, α ) θα , for θ ∈ M G ( U ) and α ∈ Aut ( U ) , the set M G ( U ) decomposes into regular Aut ( U ) -orbits M P,KG ( U ) , one for each section ( P, K ) ∈ Q G ( U ) . Asthe action of Aut ( U ) commutes with that of G , it induces an Aut ( U ) -action ([ θ ] G , α ) [ θα ] G on the set M G ( U ) /G of G -classes. This action can be used to classify the G -classesof U -morphisms as follows. Proposition 4.2.
Let U ⊑ G . (i) As Aut ( U ) -set, M G ( U ) /G is the disjoint union of transitive Aut ( U ) -sets M P,KG ( U ) /G := { [ θ ] G : θ ∈ M P,KG ( U ) } ,one for each G -class of sections [ P, K ] G ∈ Q G ( U ) /G . (ii) Let θ : P/K → U be a U -morphism of G . Then M P,KG ( U ) /G = { [ θα ] G : α ∈ D θ } ,where D θ is a transversal of the right cosets A θ α of A θ in Aut ( U ) . Note that, by an abuse of notation, M P,KG ( U ) /G is the set of full G -orbits of the U -morphisms in M P,KG ( U ) , although M P,KG ( U ) is not a G -set in general. Proof.
Let X = M G ( U ) and let Y = Q G ( U ) . Then X can be identified with the G -invariant subset Z of X × Y consisting of those pairs ( θ, ( P, K )) where θ has domain P/K .By Lemma 2.6,
Z/G is the disjoint union of Aut ( U ) -orbits M P,KG ( U ) /G , one for each G -class [ P, K ] G of sections of G . N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 15
Now let θ : P/K → U be a U -morphism. The stabilizer of the section ( P, K ) in G is itsnormalizer N G ( P, K ) . The automizer A G ( θ ) transforms this action into the subgroup A θ of Aut ( U ) . As Aut ( U ) acts regularly on M P,KG ( U ) , the A θ -orbits on this set correspondto the cosets of A θ in Aut ( U ) and { [ θ ] G : θ ∈ M P,KG ( U ) } = { [ θα ] G : α ∈ D θ } . (cid:3) As Inn ( U ) A θ for all θ ∈ M G ( U ) , the Aut ( U ) -action on M G ( U ) /G can be regardedas an Out ( U ) -action. Thus, for each section ( P, K ) of G , the set M P,KG ( U ) /G is isomorphicto Out ( U ) /O θ as Out ( U ) -set. Example 4.3.
Let G = A and U = . Then Q G ( U ) = { ( , 1 ) } , and Aut ( U ) ∼ = S makes two orbits on the U -morphisms of the form θ : /1 → U , as A θ ∼ = .4.2. Comparing Morphisms.
By Goursat’s Lemma (Lemma 2.1), a U -morphism θ : P/K → U corresponds to the subgroup L = { ( p, ( pK ) θ ) : p ∈ P } G × U .We call L the graph of θ . The partial order on the subgroups of G × U induces a naturalpartial order on M G ( U ) , as follows: if θ and θ ′ are U -morphisms with graphs L and L ′ then we define θ ′ θ : ⇐⇒ L ′ L .This partial order on M G ( U ) is closely related to the order P/K on Q G ( U ) . Proposition 4.4.
Let θ : P/K → U and θ ′ : P ′ /K ′ → U be U -morphisms of G . Then θ ′ θ ⇐⇒ ( P ′ , K ′ ) P/K ( P, K ) and θ ′ = φθ ,where φ : P ′ /K ′ → P/K is the homomorphism defined by ( pK ′ ) φ = pK for p ∈ P ′ .Proof. Let L = { ( p, ( pK ) θ ) : p ∈ P } be the graph of θ and let L ′ = { ( p, ( pK ′ ) θ ′ ) : p ∈ P ′ } be that of θ ′ .Assume first that L ′ L . This clearly implies P ′ P and K ′ K . Moreover, forany p ∈ P ′ , if ( p, ( pK ′ ) θ ′ ) ∈ L ′ L then ( pK ′ ) θ ′ = ( pK ) θ = ( pK ′ ) φθ as ( p, ( pK ) θ ) isthe unique element in L with first component p . Hence θ ′ = φθ . Now φ = θ ′ θ − is anisomorphism, whence ( P ′ , K ′ ) P/K ( P, K ) .Conversely, if ( P ′ , K ′ ) ( P, K ) and θ ′ = φθ then clearly ( p, ( pK ′ ) θ ′ ) = ( p, ( pK ′ ) φθ ) =( p, ( pK ) θ ) ∈ L for all p ∈ P ′ , whence L ′ L . (cid:3) More generally, for finite groups
U, U ′ ⊆ G , suppose that sections ( P, K ) ∈ Q G ( U ) and ( P ′ , K ′ ) ∈ Q G ( U ′ ) are such that ( P ′ , K ′ ) ( P, K ) with canonical homomorphism φ : P ′ /K ′ → P/K . If θ : P/K → U and θ ′ : P ′ /K ′ → U ′ are isomorphisms then thecomposition λ := ( θ ′ ) − φθ obviously is a homomorphism from U ′ to U , see Fig. 5. P/K UP ′ /K ′ U ′ θθ ′ φ λ Figure 5. λ : U ′ → U In case U = U ′ , the previous lemma says that θ ′ θ if and only if λ = id U . If U = U ′ then θ and θ ′ are incomparable. However, there are the following connections to thepartial orders on Q G . Lemma 4.5.
Let θ : P/K → U be a U -morphism. Then θ induces (i) an order preserving bijection between the sections ( P ′ , K ′ ) of G with ( P ′ , K ′ ) K ( P, K ) and the subgroups of U ; (ii) an order preserving bijection between the sections ( P ′ , K ′ ) of G with ( P ′ , K ′ ) > P ( P, K ) and the normal subgroups of U .Proof. This is an immediate consequence of Remark 3.9 on the Correspondence Theorem. (cid:3)
The Partial Order of Morphism Classes.
The partial order on M G ( U ) iscompatible in the sense of Section 2.5 with the conjugation action of G , and hence yieldsa class incidence matrix A GU ( ) = (cid:0) a ( θ, θ ′ ) (cid:1) [ θ ] , [ θ ′ ] ∈ M G ( U ) /G , where, for θ, θ ′ ∈ M G ( U ) , a ( θ, θ ′ ) = { θ a > θ ′ : a ∈ G } . This matrix is a submatrix of the class incidence matrix of the subgroup lattice of G × U ,corresponding to the classes of subgroups which occur as graphs of U -morphisms. Proposition 4.6.
Suppose that θ, θ ′ ∈ M G ( U ) have graphs L, L ′ G × U . Then a ( θ, θ ′ ) = { L ( a,u ) > L ′ : ( a, u ) ∈ G × U } .Proof. The result follows if we can show that the ( G × U ) -orbit of L is not larger thanits G -orbit. For this, let u ∈ U . Then ( p, u ) ∈ L for some p ∈ P and hence L ( p,u ) = L .But then L ( ) = L ( p − ,1 ) . (cid:3) As, for θ, θ ′ ∈ M G ( U ) and α ∈ Aut ( U ) , we have θ ′ θ ⇐⇒ θ ′ α θα, the matrix A GU ( ) is compatible (in the sense of Section 2.5) with the action of Out ( U ) on M G ( U ) /G . In fact, this relates it to the class incidence matrix A U ( P/K ) of Q G ( U ) as follows. Proposition 4.7.
With the row summing and column picking matrices corresponding tothe
Out ( U ) -orbits on M G ( U ) /G , we have A U ( P/K ) = R ( Out ( U )) · A GU ( ) · C ( Out ( U )) .Proof. By Proposition 4.2(i), the union of the classes [ θα ] G , α ∈ Aut ( U ) , is the set ofall U -morphisms of the form ( P/K ) a → U for some a ∈ G . This set contains, for eachconjugate ( P/K ) a with ( P/K ) a > P/K P ′ /K ′ , exactly one U -morphism above θ ′ : P ′ /K ′ → U , by Proposition 4.4. (cid:3) Example 4.8.
Let G = A and U = . Then M G ( U ) /G consists of three classes, onewith ( P, K ) = (
3, 1 ) and two with ( P, K ) = ( A , 2 ) , permuted by Out ( U ) . We have A GU ( ) = (cid:18)
11 11 · (cid:19) , A U ( P/K ) = (cid:16) · ·· (cid:17) · (cid:18)
11 11 · (cid:19) · (cid:18) ·· · · (cid:19) = (cid:18)
12 1 (cid:19) . N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 17 Subgroups of a Direct Product
From now on, let G and G be finite groups. In this section we describe the sub-groups and the conjugacy classes of subgroups of the direct product G × G in terms ofproperties of the groups G and G . By Goursat’s Lemma 2.1, the subgroups of G × G correspond to isomorphisms between sections of G and G . Any such isomorphism arisesas composition of two U -morphisms, for a suitable finite group U . This motivates thestudy of subgroups of G × G as pairs of U -morphisms.5.1. Pairs of Morphisms.
Let U be a finite group. We call L = ( θ : P /K → P /K ) a U -subgroup of G × G if U is its Goursat type, i.e., if P i /K i ∼ = U , i =
1, 2 , and we denoteby S G × G ( U ) the set of all U -subgroups of G × G . Given morphisms θ i : P i /K i → U in M G i ( U ) , i =
1, 2 , composition yields an isomorphism θ = θ θ − : P /K → P /K withwhose graph is a U -subgroup L G × G . Hence there is a map Π : M G ( U ) × M G ( U ) → S G × G ( U ) defined by Π ( θ , θ ) = θ θ − . In fact, the ( G , G ) -biset S G × G ( U ) is the tensor product of the ( G , Aut ( U )) -biset M G ( U ) and the opposite of the ( G , Aut ( U )) -biset M G ( U ) . Proposition 5.1. S G × G ( U ) = M G ( U ) × Aut ( U ) M G ( U ) op .Proof. For any U -subgroup L = ( θ : P /K → P /K ) there exist θ i ∈ M G i ( U ) , i =
1, 2 ,such that θ = Π ( θ , θ ) . Moreover, for θ i , θ ′ i ∈ M G i ( U ) , we have Π ( θ , θ ) = Π ( θ ′ , θ ′ ) if and only if θ ′ θ − = θ ′ θ − in Aut ( U ) . (cid:3) It will be convenient to express the order of a U -subgroup in terms of U . Lemma 5.2.
Let L = ( θ : P /K → P /K ) be a U -subgroup of G × G . Then | L | = | K || K || U | = | P || P | / | U | . Comparing Subgroups.
Let U , U ′ be finite groups. We now describe and analyzethe partial order of subgroups of G × G in terms of pairs of morphisms. Proposition 5.3.
Let ( θ i : P i /K i → U ) ∈ M G i ( U ) and ( θ ′ i : P ′ i /K ′ i → U ′ ) ∈ M G i ( U ′ ) , i =
1, 2 ,be morphisms, let θ = Π ( θ , θ ) , θ ′ = Π ( θ ′ , θ ′ ) with corresponding subgroups L , L ′ of G × G . Then L ′ L if and only if (i) ( P ′ i , K ′ i ) ( P i , K i ) as sections of G i , i =
1, 2 ; and (ii) λ = λ , where λ i = ( θ ′ i ) − φ i θ i , and φ i : P ′ i /K ′ i → P i /K i is the homomorphismdefined by ( K ′ i p ) φ i = K i p , for p ∈ P ′ i , i =
1, 2 . P /K UP ′ /K ′ U ′ P /K U P ′ /K ′ U ′ θ θ ′ φ λ θ θ ′ φ λ Figure 6. L ′ L Proof.
Write L ′ = { ( p ′ , p ′ ) ∈ P ′ × P ′ : ( p ′ K ′ ) θ ′ = ( p ′ K ′ ) θ ′ } and L = { ( p , p ) ∈ P × P : ( p K ) θ = ( p K ) θ } .Then L ′ L if and only if ( P ′ i , K ′ i ) ( P i , K i ) , i =
1, 2 , and, for p i ∈ P ′ i , we have ( p K ) θ = ( p K ) θ . But if p i ∈ P ′ i then ( p i K i ) θ i = ( p i K ′ i ) φ i θ i = ( p i K ′ i ) θ ′ i λ i .So ( p K ) θ = ( p K ) θ if and only if λ = λ , see Fig. 6. (cid:3) Corollary 5.4.
With the notation of Proposition 5.3, L ′ L if and only if (i) ( P ′ i , K ′ i ) ( P i , K i ) as sections of G i , i =
1, 2 ; (ii) φ θ = θ ′ φ The partial orders on sections introduced in Definition 3.7 give rise to relations on thesubgroups of G × G , as follows. Definition 5.5.
Let L = ( θ : P /K → P /K ) and L ′ = ( θ ′ : P ′ /K ′ → P ′ /K ′ ) besubgroups of G × G and suppose that L ′ L . We write(i) L ′ P L , if ( P ′ i , K ′ i ) P ( P i , K i ) , i =
1, 2 ,i.e., if both sections of L ′ and L have the same top groups;(ii) L ′ K L , if ( P ′ i , K ′ i ) K ( P i , K i ) , i =
1, 2 ,i.e., if both sections of L ′ and L have the same bottom groups;(iii) L ′ P/K L , if ( P ′ i , K ′ i ) P/K ( P i , K i ) , i =
1, 2 ,i.e., if the canonical homomorphisms φ i : P ′ i /K ′ i → P i /K i are isomorphisms.All three relations are obviously partial orders. Moreover, they decompose the partialorder on the subgroups of G × G , in analogy to Corollary 3.8. Theorem 5.6.
Let L = ( θ : P /K → P /K ) and L ′ = ( θ ′ : P ′ /K ′ → P ′ /K ′ ) be such that L ′ L . Define a map ˆ θ ′ : P ′ / ( P ′ ∩ K ) → P ′ / ( P ′ ∩ K ) by ( p ( P ′ ∩ K )) ˆ θ ′ = p ( P ′ ∩ K ) ,whenever p i ∈ P ′ i are such that ( p P ′ ) θ ′ = p P ′ , and a map ˜ θ : P ′ K /K → P ′ K /K by ( p K ) ˜ θ = p K whenever p i ∈ P ′ i are such that ( p K ) θ = p K . Then (i) ˆ θ ′ and ˜ θ are isomorphisms with corresponding graphs L ˆ θ ′ and L ˜ θ G × G . (ii) L ˆ θ ′ and L ˜ θ are the unique subgroups of G × G with L ′ P L ˆ θ ′ P/K L ˜ θ K L .Proof. Denote by φ i : P ′ i /K ′ i → P i /K i the canonical homomorphism, i =
1, 2 . Then,as in the proof of Theorem 3.6, φ i is the product of an epimorphism φ i1 : P ′ i /K ′ i → ( P ′ i /K ′ i ) / ker φ i , an isomorphism φ i2 : ( P ′ i /K ′ i ) / ker φ i → im φ i , and a monomorphism φ i3 : im φ i → P i /K i . By Corollary 5.4, φ θ = θ ′ φ . It follows that ( im φ ) θ = im φ and ( ker φ ) θ ′ = ker φ . Thus θ restricts to an isomorphism ˜ θ from im φ to im φ , and θ ′ induces an isomorphism ˆ θ ′ from ( P ′ /K ′ ) / ker φ to ( P ′ /K ′ ) / ker φ , and the followingdiagram commutes.By Proposition 3.6, im φ i = P ′ i K i /K i and ( P ′ i /K ′ i ) / ker φ i ∼ = P ′ i / ( P ′ i ∩ K i ) . (cid:3) Corollary 5.7.
The partial order on S G × G is a product of three relations: = K ◦ P/K ◦ P .Moreover, if A ( R ) denotes the incidence matrix of the relation R , the stronger property A ( ) = A ( K ) · A ( P/K ) · A ( P ) also holds.Proof. Like Corollary 3.8, this follows from the uniqueness of the intermediate subgroupsin Theorem 5.6. (cid:3)
N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 19 P /K P /K ( P ′ /K ′ ) / ker φ ( P ′ /K ′ ) / ker φ P ′ /K ′ P ′ /K ′ im φ im φ φ φ θθ ′ ˆ θ ′ ˜ θφ φ φ φ φ φ Figure 7. θ ′ ˆ θ ′ ˜ θ θ Lemma 5.8.
Let ( θ i : P i /K i → U ) ∈ M G i ( U ) , i =
1, 2 and L = Π ( θ , θ ) . Then (i) the set { L ′ G × G : L ′ K L } is in an order preserving bijective correspondencewith the subgroups of U ; (ii) the set { L ′ G × G : L P L ′ } is in an order preserving bijective correspondencewith the quotients of U ; (iii) the set { L ′ : L ′ P/K L } is in an order preserving bijective correspondence with { ( P ′ , K ′ ) : ( P ′ , K ′ ) P/K ( P , K ) } × { ( P ′ , K ′ ) : ( P ′ , K ′ ) P/K ( P , K ) } ; (iv) the set { L ′ : L P/K L ′ } is in an order preserving bijective correspondence with { ( P ′ , K ′ ) : ( P ′ , K ′ ) > P/K ( P , K ) } × { ( P ′ , K ′ ) : ( P ′ , K ′ ) > P/K ( P , K ) } .Proof. This follows from Lemma 4.5 on the correspondences induced by a U -morphism,together with Proposition 4.4 and Theorem 5.6. (cid:3) Classes of Subgroups.
The conjugacy classes of U -subgroups of G × G can bedescribed as Aut ( U ) -orbits of pairs of classes of U -morphisms. Theorem 5.9.
Let U ⊑ G i , i =
1, 2 . (i) S G × G ( U ) / ( G × G ) is the disjoint union of sets M P ,K G ( U ) /G × Aut ( U ) ( M P ,K G ( U ) /G ) op , one for each pair of section classes [ P i , K i ] G i ∈ Q G i ( U ) /G i . (ii) Let θ i : P i /K i → U be U -morphisms of G i , i =
1, 2 . Then M P ,K G ( U ) /G × Aut ( U ) ( M P ,K G ( U ) /G ) op = { [ θ dθ − ] G × G : d ∈ D θ ,θ } ,where D θ ,θ is a transversal of the ( A θ , A θ ) -double cosets in Aut ( U ) .Proof. (i) As S G × G ( U ) = M G ( U ) × Aut ( U ) M G ( U ) , the ( G × G ) -conjugacy classesof U -subgroups of G × G are Aut ( U ) -orbits on the direct product M G ( U ) /G × M G ( U ) /G . By Proposition 4.2(i), this direct product is the disjoint union of Aut ( U ) -invariant direct products M P ,K G ( U ) /G × M P ,K G ( U ) /G , one for each choice of G i -classesof sections [ P i , K i ] G i ∈ Q G i ( U ) /G i , i =
1, 2 .(ii) Let α i ∈ Aut ( U ) , i =
1, 2 . Note first that the image of [ θ α ] G × [ θ α ] G under Π is a ( G × G ) -conjugacy class of U -subgroups and that each ( G × G ) -class is of thisform. We show that the classes in M P ,K G ( U ) /G × M P ,K G ( U ) /G correspond to the ( A θ , A θ ) -double cosets in Aut ( U ) . For this, let α ′ i ∈ Aut ( U ) , i =
1, 2 , and assume that Π ([ θ α ] G , [ θ α ] G ) = Π ([ θ α ′ ] G , [ θ α ′ ] G ) . By Proposition 4.2(ii), this is the case ifand only if θ A θ α α − A θ θ − = θ A θ α ′ ( α ′ ) − A θ θ − , i.e., if α α − and α ′ ( α ′ ) − lie in the same ( A θ , A θ ) -double coset. (cid:3) Example 5.10.
Let G = S . For each U -morphism θ : P/K → U , we have O θ = Out ( U ) .Therefore, by Theorem 5.9, there exists exactly one conjugacy class of subgroups for eachpair of classes of isomorphic sections ( P , K ) , ( P , K ) . A transversal { L , . . . , L } of the conjugacy classes of subgroups of G × G can be labelled by pairs of sections as follows. (
1, 1 ) (
2, 2 ) (
3, 3 ) (
G, G )(
1, 1 ) L L L L (
2, 2 ) L L L L (
3, 3 ) L L L L ( G, G ) L L L L (
2, 1 ) (
G, 3 ) (
3, 1 ) (
G, 1 )(
2, 1 ) L L ( G, 3 ) L L (
3, 1 ) L ( G, 1 ) L Here, a subgroup L i in row ( P , K ) and column ( P , K ) has a Goursat isomorphism ofthe form P /K → P /K .The normalizer of a subgroup θ = Π ( θ , θ ) of G × G , described as a quotientof two U -morphisms θ i , can be described as the quotient of the automizers of the two U -morphisms. Theorem 5.11.
Let U ⊑ G i and let θ i ∈ M G i ( U ) , for i =
1, 2 . Then N G × G ( Π ( θ , θ )) = A G ( θ ) ∗ A G ( θ ) op Proof.
For i =
1, 2 , suppose that θ i : P i /K i → U and let ( ˜ P i , ˜ K i ) = A G i ( P i , K i ) . Then A G i ( θ i ) : ˜ P i / ˜ K i → A θ i Aut ( U ) is the automizer of θ i . Let θ = Π ( θ , θ ) = θ θ − .Then, on the one hand, N G × G ( θ ) = { ( a , a ) ∈ G × G : γ − θγ a = θ } consists of those elements ( a , a ) ∈ ˜ P × ˜ P which induce automorphisms α i = θ − γ a i θ i =( a i ˜ K i ) A Gi ( θ i ) ∈ Aut ( U ) such that θ α − α θ − = θ θ − , i.e., α = α .On the other hand, by the Lemma 2.3, A G ( θ ) ∗ A G ( θ ) op = Π ( ˜ θ ′ ( ˜ θ ′ )) where, for i =
1, 2 , ˜ θ ′ i : ˜ P ′ i / ˜ K i → ˜ U is the restriction of the isomorphism A G i ( θ i ) to the preimage ˜ P ′ i / ˜ K i of ˜ U = A θ ∩ A θ in˜ P i / ˜ K i . Hence, as a subgroup of G × G , the product A G ( θ ) ∗ A G ( θ ) op consists ofthose elements ( a , a ) ∈ ˜ P ′ × ˜ P ′ with ( a ˜ K ) A G1 ( θ ) = ( a ˜ K ) A G2 ( θ ) .It follows that A G ( θ ) ∗ A G ( θ ) op = N G × G ( θ ) , as desired. (cid:3) As an immediate consequence, we can determine the normalizer index of a subgroupof G × G in terms of U -morphisms. Corollary 5.12.
Let L = Π ( θ , θ ) G × G , for θ i : P i /K i → U , i =
1, 2 . Then | N G × G ( L ) : L | = | C N ( P ) | | C N ( P ) | | O θ ∩ O θ | | Z ( U ) | − ,where N i = N G i ( K i ) /K i and P i = P i /K i , i =
1, 2 .Proof.
By Lemma 5.2, | L | = | K | | K | | U | . With the notation from the preceding proof, | N G × G ( L ) | = | ˜ K | | ˜ K | | ˜ U | . Thus | N G × G ( L ) : L | = | ˜ K | | ˜ K | | ˜ U || K | | K | | U | . N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 21
But | ˜ K i : K i | = | C N i ( P i ) | , i =
1, 2 , by Definition 3.2. Moreover, | ˜ U | = | A θ ∩ A θ | = | Inn ( U ) | | O θ ∩ O θ | and | U | = | Inn ( U ) | | Z ( U ) | . (cid:3) Table of Marks
We are now in a position to assemble the table of marks of G × G from a collectionof smaller class incidence matrices. Theorem 6.1.
Let G and G be finite groups. Then the table of marks of G × G is M ( G × G ) = D · A ( K ) · A ( P/K ) · A ( P ) , where D is the diagonal matrix with entries | N G × G ( L ) : L | , for L running over atransversal of the conjugacy classes of subgroups of G × G .Proof. The proof is similar to that of Theorem 3.10 in combination with Corollary 5.4. (cid:3)
In the remainder of this section, we determine the block diagonal structure of each ofthe matrices A ( K ) , A ( P/K ) and A ( P ) of G × G .6.1. The class incidence matrix of the ( G × G ) -poset ( S G × G , K ) is a block diagonalmatrix, with one block for each pair ([ K ] , [ K ]) of conjugacy classes [ K i ] of subgroups of G i , i =
1, 2 . Theorem 6.2.
For K i G i , i =
1, 2 , denote by A K ,K the class incidence matrix of N G ( K ) × N G ( K ) acting on the subposet of ( S G × G , ) consisting of those subgroups L with bottom groups k i ( L ) = K i , i =
1, 2 . Then A ( K ) = M [ K i ] ∈ S Gi /G i i = A K ,K .Proof. Let X = S G × G and Y = S G × S G . We identify X with Z ⊆ X × Y , where Z := { ( x, y ) : ( k ( x ) , k ( x )) = y } . Then Lemma 2.6 yields a partition of the conjugacyclasses of subgroups of G × G indexed by [ K i ] ∈ S G i /G i , i =
1, 2 . The stabilizer of y = ( K , K ) ∈ Y is N G ( K ) × N G ( K ) and Zy = { x ∈ X : ( k ( x ) , k ( x )) = y } .Let L G × G be such that k i ( L ) = K i , i =
1, 2 . In order to count the ( G × G ) -conjugates of a subgroup L ′ G × G with bottom groups k i ( L ′ ) = K i , i =
1, 2 , above L in the K -order, it suffices to note that L K ( L ′ ) g for some g ∈ G × G if and onlyif L K ( L ′ ) g for some g ∈ N G ( K ) × N G ( K ) .Finally by the definition of K there are no incidences between subgroups with different K i , giving the block diagonal structure. (cid:3) Example 6.3.
Let G = G = S . Then A ( K ) is the block sum of the matrices A K ,K in the table below, with rows and columns labelled by the conjugacy classes of subgroups K of S . Within A K ,K , the row label of a subgroup of the form P /K → P /K is just P → P , for brevity. The column labels are identical and have been omitted. K 1 2 3 S → · · · → · · → · · S → S → → · → S → S → → → → S → · S → → → · S → S → S S → → → → S ( G × G ) -poset ( S G × G , P/K ) is a block diagonalmatrix, with one block for each group U ⊆ G i , i =
1, 2 , up to isomorphism.
Definition 6.4.
For a finite group G and finite G -sets X and X , let A i be a squarematrix with rows and columns labelled by X i , i =
1, 2 . The action of G on X × X permutes the rows and columns of the Kronecker product A ⊗ A . If the matrices A and A are compatible with the G -action then so is their Kronecker product, and wedefine A ⊗ G A := R ( G ) · ( A ⊗ A ) · C ( G ) ,where the row summing and column picking matrices R ( G ) and C ( G ) have been con-structed as in Lemma 2.7, with respect to the G -orbits on X × X .For U ⊑ G i , consider the class incidence matrices A i = A G i U ( ) of the G i -posets M G i ( U ) , i =
1, 2 , from Section 4.3. By Proposition 4.7, these matrices, and hence A ⊗ A , are compatible with the action of Out ( U ) on their rows and columns. Theorem 6.5.
We have A ( P/K ) = M U ⊑ G ,G A G U ( ) ⊗ Out ( U ) A G U ( ) ,where, for U ⊑ G i , A G i U ( ) is the class incidence matrix of the G i -poset M G i ( U ) , i =
1, 2 .Proof.
Let L ′ = ( θ ′ : P ′ /K ′ → P ′ /K ′ ) be a subgroup of G × G with Goursat type U andselect U -morphisms θ ′ and θ ′ such that Π ( θ ′ , θ ′ ) = θ ′ . By Lemma 5.8 (iv) the subgroups L of G × G with L > P/K L ′ correspond to pairs of sections ( P i , K i ) > P/K ( P ′ i , K ′ i ) , i =
1, 2 . For each such section ( P i , K i ) , set θ i = φ − θ ′ i , where φ i : P ′ i /K ′ i → P i /K i is thecanonical isomorphism. By Proposition 4.4, θ i : P i /K i → U is the unique U -morphismin M P i ,K i G i ( U ) with θ ′ i θ i . The number of conjugates L x of a subgroup L of G × G with L x > P/K L ′ is thus equal to the number of pairs ( θ , θ ) ∈ M G ( U ) × M G ( U ) with θ ′ i θ i such that Π ( θ , θ ) is a conjugate of L in G × G .If L = Π ( θ , θ ) for θ i ∈ M P i ,K i G i ( U ) then, by Theorem 5.9, the set of all pairs of U -morphisms mapping to a conjugate of L under Π is the Out ( U ) -orbit of [ θ ] G × [ θ ] G in M P ,K G ( U ) /G × M P ,K G ( U ) /G . The number of G i -conjugates of θ i above θ ′ i is given bythe entry a ( θ i , θ ′ i ) of the class incidence matrix A G i U . By Proposition 4.2, the Aut ( U ) -set M P i ,K i G i ( U ) /G i is isomorphic to Aut ( U ) /A θ i . Hence { L x > P/K L ′ : x ∈ G × G } = X α ∈ T θ1,θ2 a ( θ α, θ ′ ) a ( θ α, θ ′ ) ,where T θ ,θ is a transversal of the right cosets of A θ ∩ A θ in Aut ( U ) . As T θ ,θ can alsobe used to represent the right cosets of O θ ∩ O θ in Out ( U ) , the same number appearsas the L, L ′ -entry of the matrix A G U ( ) ⊗ Out ( U ) A G U ( ) . (cid:3) Example 6.6.
Let G = G = S . Then A ( P/K ) is the block sum of the followingmatrices A G U ( ) ⊗ Out ( U ) A G U ( ) . As Out ( U ) here acts trivially, the matrices are simplythe Kronecker squares of the matrices A GU ( ) in Example 3.16. The column labels are N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 23 identical to the row labels and have been omitted. U A G U ( ) ⊗ Out ( U ) A G U ( ) → · · · · · · · · · · · · · · · → · · · · · · · · · · · · · · → · · · · · · · · · · · · · · → S /S · · · · · · · · · · · · → · · · · · · · · · · · · · · → · · · · · · · · · · · · → · · · · · · · · · · · · → S /S · · · · · · · · → · · · · · · · · · · · · · · → · · · · · · · · · · · · → · · · · · · · · · · · · → S /S · · · · · · · · S /S → · · · · · · · · · · · · S /S → · · · · · · · · S /S → · · · · · · · · S /S → S /S → · · · → S /3 1 1 · · S /3 → · · S /3 → S /3 1 1 1 1 → S /1 → S /1 1 Example 6.7.
Continuing Example 4.8 for G = A and U = , we have A GU ( ) = (cid:18)
11 11 · (cid:19) , A GU ( ) ⊗ Out ( U ) A GU ( ) =
12 12 ·
12 1 1 12 1 1 · , illustrating the effect of a non-trivial Out ( U ) -action.6.3. The class incidence matrix of the ( G × G ) -poset ( S G × G , P ) is a block diagonalmatrix, with one block for each pair ([ P ] , [ P ]) of conjugacy classes [ P i ] of subgroups of G i , i =
1, 2 . Theorem 6.8.
For P i G i , i =
1, 2 , denote by A P ,P the class incidence matrix of N G ( P ) × N G ( P ) acting on the sub poset of ( S G × G , ) consisting of those subgroups L with p i ( L ) = P i , i =
1, 2 . Then A ( P ) = M P i ∈ S Gi /G i ,i = A P ,P ,Proof. Similar to the proof of Theorem 6.2, with X = S G × G , Y = S G × S G Z = { ( x, y ) :( p ( x ) , p ( x )) = y } ⊆ X × Y . (cid:3) Example 6.9.
Again we let G = G = S . Then A ( P ) is the block sum of the matrices A P ,P in the table below, with rows and columns labelled by the conjugacy classes ofsubgroups P of S . Similar to Example 6.3, within A P ,P , the row label of a subgroup ofthe form P /K → P /K is just K → K , for brevity. The column labels are identicaland have been omitted. P 1 2 3 S → → → → S → → · → → → · → S → → → · → → S S → → · S → → → · · → · S → S Example 6.10.
Combining the matrices from Examples 6.3, 6.6 and 6.9 according toTheorem 6.1, yields the table of marks M of S × S with rows and columns sorted bysection size, as in Example 6.6: M = · · · · · · · · · · · · · · · · · · · · ·
18 6 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · The Double Burnside Algebra of S As an application of the ideas from previous sections we now construct a mark homo-morphism for the rational double Burnside algebra of G = S .7.1. The Double Burnside Ring.
Let G , H and K be finite groups. The Grothendieckgroup of the category of finite ( G, H ) -bisets is denoted by B ( G, H ) . If ( G, H ) -bisets areidentified with G × H -sets, then the abelian group B ( G, H ) is identified with the Burnsidegroup B ( G × H ) , and hence the transitive bisets [ G × H/L ] , where L runs through atransversal of the conjugacy classes of subgroups of G × H , form a Z -basis of B ( G, H ) .There is a bi-additive map from B ( G, H ) × B ( H, K ) to B ( G, K ) given by ([ X ] , [ Y ]) [ X ] · H [ Y ] = [ X × H Y ] .Multiplication of transitive bisets is described by the following Mackey-formula. Proposition 7.1 ([3, 2.3.24]) . Let L G × H and M H × K . Let X ⊆ H be a transversalof the ( p ( L ) , p ( M )) -double cosets in H . Then [( G × H ) /L ] · H [( H × K ) /M ] = X x ∈ X [( G × K ) / ( L ( ) ∗ M )] With this multiplication, in particular, B ( G, G ) is a ring, the double Burnside ring of G .The rational double Burnside algebra Q B ( G, G ) = Q ⊗ Z B ( G, G ) is known to besemisimple, if and only if G is cyclic [3, Proposition 6.1.7]. Little more is known aboutthe structure of Q B ( G, G ) in general. N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 25
A Mark Homomorphism for the Double Burnside Ring of S . For the ordi-nary Burnside ring B ( G ) , the table of marks of G is the matrix of the mark isomorphism β G : Q B ( G ) → Q r between the rational Burnside algebra and its ghost algebra. It is anopen question, whether there exist equivalent constructions of ghost algebras and markhomomorphims for the double Burnside ring. Boltje and Danz [2] have investigated therole of the table of marks of the direct product G × G in this context. Here, we usethe decomposition of the table of marks of G × G from Theorem 6.1 and the idea oftransposing the P part from Section 3.5 in order to build a satisfying ghost algebra forthe group G = S .For this purpose, we first set up a labelling of the natural basis of Q B ( G, G ) as fol-lows. Set I = {
1, . . . , 22 } . Let { L i : i ∈ I } be the conjugacy class representatives fromExample 5.10. Then the rational Burnside algebra Q B ( G, G ) has a Q -basis consisting ofelements b i = [ G × G/L i ] , i ∈ I , and multiplication defined by 7.1.By Theorem 6.1, the table of marks M of G × G is a matrix product M = D · A ( K ) · A ( P/K ) · A ( P ) ,of a diagonal matrix D with entries | N G × G ( L i ) : L i | , i ∈ I , and three class incidencematrices. For our purpose, we now modify this product and set M ′ = D · A ( K ) · A ( P/K ) · D · A ( > P ) · D ,where D = diag (
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 6, 6, 6, 6, 1, 1, 6, 6, 1, 1 ) , D = diag (
1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 6 ) ,are diagonal matrices. The resulting matrix is M ′ = / / . . / / . . . . . . . . . . . . . . . .1 . 1 . 1 . 1 . . . . . . . . . . . . . . . / / / / / / / / . . . . . . . . . . . . . .2 . . . . . . . 4 . . . . . . . . . . . . .1 / . . . . . . 2 / . . . . . . . . . . . . / . / . . . . . / . / . . . . . . . . . . . / / / / . . . . / / / / . . . . . . . . . .1 . . . 3 . . . 2 . . . 6 . . . . . . . . . / / . . / / . . 1 / . . 3 1 . . . . . . . . / . / . 1 . 1 . / . / . 2 . 2 . . . . . . . / / / / / / / / / / / / / . / . . 1 . 1 / . / . . 2 . 2 1 1 2 2 . .2 . . . . . . . . . 2 . . . . . . . . . 2 .1 . . . . 1 . . . . 1 . . . . . 1 . . . 1 1 The matrix M ′ = ( m ′ ij ) is obviously invertible, hence there are unique elements c j ∈ Q B ( G, G ) , j ∈ I , such that b i = X j ∈ I m ′ ij c j ,forming a new Q -basis of Q B ( G, G ) . Theorem 7.2.
Let G = S . Then the linear map β ′ G × G : Q B ( G, G ) → Q × defined by β ′ G × G (cid:16) X i ∈ I x i c i (cid:17) = x x x x . . . .x x x x . . . .x x x x . . . .. . . x . . . .. . . . x x . .. . . . . x . .. . . . . . x .x x x x x x . x ,where c i ∈ Q B ( G, G ) are defined as above, and x i ∈ Q , i ∈ I , is an injective homomor-phism of algebras.Proof. This claim is validated by an explicit calculation, whose details we omit. Thegeneral strategy is as follows. For i ∈ I , let C i be the matrix of c i in the right regularrepresentation of Q B ( G, G ) (computed with the help of the Mackey formula in Proposi-tion 7.1). Let ≡ be the equivalence relation on I corresponding to the kernel of the mapthat sends the conjugacy class of a subgroup L = ( P/K → P ′ /K ′ ) to the conjugacy classof the section P ′ /K ′ . Then ≡ partitions I as {{
1, 5, 9, 13 } , {
2, 6, 10, 14 } , {
3, 7, 11, 15 } , {
4, 8, 12, 16 } , {
17, 19 } , {
18, 20 } , { } , { }} .It turns out that all transposed matrices C Ti are compatible with the equivalence ≡ in thesense of Section 2.5. Hence, after choosing a transversal of ≡ , and using the correspondingrow summing and column picking matrices R ( ≡ ) and C ( ≡ ) , the map β ′ defined by β ′ ( c i ) = C ( ≡ ) T · C i · R ( ≡ ) T , i ∈ I ,is independent of the choice of transversal. In fact, β ′ = β ′ G × G . By Lemma 2.7, β ′ ( c i c k ) = C ( ≡ ) T · C i · C k · R ( ≡ ) T = β ′ ( c i ) · C ( ≡ ) T · C k · R ( ≡ ) T = β ′ ( c i ) · β ′ ( c k ) ,for i, k ∈ I , showing that β ′ G × G = β ′ is a homomorphism. Injectivity follows from adimension count. (cid:3) It might be worth pointing out that the equivalence ≡ , and hence the notion of com-patibility and the map β ′ depend on the basis used for the matrices of the right regularrepresentation. In the case G = S , the natural basis { b i } of Q B ( G, G ) also yields com-patible matrices, but the corresponding map β ′ is not injective. A base change underthe table of marks of G × G gives matrices which are not compatible. Changing basisunder the matrix product D · A ( K ) · A ( P/K ) · A ( > P ) yields compatible matrices andan injective homomorphism like M ′ does. Our matrices β ′ G × G ( c i ) have the added ben-efit of being normalized and extremely sparse, exposing other representation theoreticproperties of the algebra Q B ( G, G ) , such as the following. Corollary 7.3.
Let G = S and denote by J the Jacobsen radical of the rational Burnsidealgebra Q B ( G, G ) . (i) With c i as above, { c i : i =
4, 8, 12, 13, 14, 15, 16, 18, 19, 20 } is a basis of J . (ii) Q B ( G, G ) /J ∼ = Q × ⊕ Q ⊕ Q ⊕ Q . The map β ′ G × G : Q B ( G, G ) → Q × can be regarded as a mark homomorphism forthe double Burnside ring of G = S . It assigns to each ( G, G ) -biset a square matrix ofrational marks. For example, for b = [( G × G ) /L ]= c + c + c + c + c + c + + + c + c + + N THE TABLE OF MARKS OF A DIRECT PRODUCT OF FINITE GROUPS. 27 we have β ′ G × G ( b ) = / . / . . . . .. 1 . 1 . . . . / . / . . . . .. . . . . . . .. . . . 1 1 . .. . . . . . . .. . . . . . . .. 2 . 2 2 2 . . , and the image of b = [ G × G/G ] = c + c + c + c + c + c is the identity matrix.While the case G = S provides only a small example, and the above constructioninvolves some ad hoc measures, we expect that for many if not all finite groups G a markhomomorphism for the rational double Burnside algebra Q B ( G, G ) can be constructed ina similar way. This will be the subject of future research. References [1] Robert Boltje and Susanne Danz,
A ghost ring for the left-free double Burnside ring and an appli-cation to fusion systems , Adv. Math. (2012), no. 3, 1688–1733. MR 2871154[2] ,
A ghost algebra of the double Burnside algebra in characteristic zero , J. Pure and Appl.Algebra (2013), no. 4, 608–635. MR 2983839[3] Serge Bouc,
Biset functors for finite groups , Lecture Notes in Mathematics, vol. 1990, Springer-Verlag, Berlin, 2010. MR 2598185[4] Serge Bouc, Radu Stancu, and Jacques Th´evenaz,
Simple biset functors and double Burnside ring ,J. Pure Appl. Algebra (2013), no. 3, 546–566. MR 2974230[5] William Burnside,
Theory of groups of finite order , 2 ed., Cambridge University Press, Cambridge,1911. MR 0069818[6] Andreas Dress,
A characterisation of solvable groups , Math. Z. (1969), no. 3, 213 – 217.MR 0248239[7] ´Edouard Goursat,
Sur les substitutions orthogonales et les divisions r´eguli`eres de l’espace , Ann. Sci.´Ecole Norm. Sup. (1889), 9–102.[8] Bertram Huppert, Endliche Gruppen I , Die Grundlehren der Mathematischen Wissenschaften, Band134, Springer-Verlag, Berlin-New York, 1967. MR 0224703[9] Joachim Lambek,
Goursat’s theorem and the Zassenhaus lemma , Canad. J. Math. (1958), 45–56.MR 0098138[10] Klaus Lux and Herbert Pahlings, Representations of groups: A computational approach , CambridgeUniversity Press, Cambridge, 2010. MR 2680716[11] Brendan Masterson,
On the table of marks of a direct product of finite groups , Ph.D. thesis, NationalUniversity of Ireland, Galway, 2016.[12] Liam Naughton and G¨otz Pfeiffer,
Computing the table of marks of a cyclic extension , Math. Comp. (2012), no. 280, 2419–2438. MR 2945164[13] G¨otz Pfeiffer, The subgroups of M or How to compute the table of marks of a finite group , Exper-iment. Math. (1997), no. 3, 247–270. MR 1481593[14] K´ari Ragnarsson and Radu Stancu, Saturated fusion systems as idempotents in the double Burnsidering , Geom. Topol. (2013), no. 2, 839–904. MR 3070516[15] Roland Schmidt, Untergruppenverb¨ande direkter Produkte von Gruppen , Arch. Math. (Basel) (1978), no. 3, 229–235. MR 0491982[16] Giovanni Zacher, On the lattice of subgroups of the Cartesian square of a simple group , Rend. Sem.Mat. Univ. Padova (1981), 235–241. MR 636640 B.M.: Department of Design Engineering and Mathematics, Middlesex University,London, The Boroughs, London NW4 4BT, United Kingdom
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