The ambiguity index of an equipped finite group
aa r X i v : . [ m a t h . AG ] A p r THE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP
F.A. BOGOMOLOV AND VIK.S. KULIKOV
Abstract.
In [10], the ambiguity index a ( G,O ) was introduced for each equippedfinite group ( G, O ). It is equal to the number of connected components of a Hurwitzspace parametrizing coverings of a projective line with Galois group G assuming thatall local monodromies belong to conjugacy classes O in G and the number of branchpoints is greater than some constant. We prove in this article that the ambiguityindex can be identified with the size of a generalization of so called Bogomolovmultiplier ([8], see also [1]) and hence can be easily computed for many pairs ( G, O ). Introduction
Let G be a finite group and O be a subset of G consisting of conjugacy classes C i of G , O = C ∪· · ·∪ C m , which together generate G . The pair ( G, O ) is called an equippedgroup and O is called an equipment of G . We fix the numbering of conjugacy classescontained in O . One can associate a C -group ( e G, e O ) to each equipped group ( G, O ).The C -group e G is generated by the letters of the alphabet Y = Y O = { y g | g ∈ O } subject to relations: y g y g = y g y g − g g = y g g g − y g . (1)We assume e O = Y O in the definition of e G .There is an obvious natural homomorphism β : e G → G given by β ( y g ) = g . Itwas shown in [10], that the commutator subgroup [ e G, e G ] is finite. The order a ( G,O ) ofthe group ker β ∩ [ e G, e G ] was called the ambiguity index of the equipped finite group( G, O ).The notion of equipped groups is related to the description of Hurwitz spacesparametrizing maps between projective curves with G as the monodromy group andthe ambiguity index a G,O is equal to the properly defined ”asymptotic” number ofconnected components of Hurwitz space parametrizing covering of curves with fixedramification data. More precisely, let f : E → F be a morphism of a non-singularcomplex irreducible projective curve E onto a non-singular projective curve F . Letus choose a point z ∈ F such that z is not a branch point of f hence the points f − ( z ) = { w , . . . , w d } , where d = deg f , are simple. If we fix the numbering ofpoints in f − ( z ) then we call f a marked covering . The second author was partially supported by grants of NSh-2998.2014.1, RFBR 14-01-00160, andboth authors were supported by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.
Let B = { z , . . . , z n } ⊂ F be the set of branch points of f . The numbering of thepoints of f − ( z ) defines a homomorphism f ∗ : π ( F \ B, z ) → Σ d of the fundamentalgroup π = π ( F \ B, z ) to the symmetric group Σ d . Define G ⊂ Σ d as im f ∗ = G .It acts transitively on f − ( z ). Let γ , . . . , γ n be simple loops around, respectively,the points z , . . . , z n starting at z . The image g j = f ∗ ( γ j ) ∈ G is called a localmonodromy of f at the point z j . Each local monodromy g j depends on the choice of γ j , therefore it is defined uniquely up to conjugation in G .Denote by O = C ∪ · · · ∪ C m ⊂ G the union of conjugacy classes of all localmonodromies and by τ i the number of local monodromies of f belonging to theconjugacy class C i . The collection τ = ( τ C , . . . , τ m C m ) is called the monodromytype of f . Assume that the elements of O generate the group G . Then the pair( G, O ) is an equipped group.Let HUR md,G,O,τ ( F, z ) be the Hurwitz space (see the definition of Hurwitz spacesin [4] or in [9]) of marked degree d coverings of F with Galois group G ⊂ Σ d , localmonodromies in O , and monodromy type τ . Hurwitz space HUR md,G,O,τ ( F, z ) mayconsists of a different number of connected components. However it was proved in[9] that for each equipped finite group ( G, O ), O = C ∪ · · · ∪ C m , there is a number T such that the number of irreducible components of each non-empty Hurwitz spaceHUR md,G,O,τ ( F, z ) is equal to a ( G,O ) if τ i > T for all i = 1 , . . . , m . The number T doesnot depend on the base curve F and degree d of the coverings.The subgroup B ( G ) ⊂ H ( G, Q/Z ) was defined and studied in [1]. It consistsof elements of H ( G, Q/Z ) which restrict trivially onto abelian subgroups of G . Itwas conjectured in [2] that B ( G ) is trivial for simple groups. This conjecture waspartially solved already in [2] and it was completely solved by Kunyavski in [8], and byKunyavski-Kang in [7] for a wider class of almost simple groups. The latter consistsof groups G which contain some simple group L and in turn are contained in theautomorphism group AutL . Kunyavski in [8] called B ( G ) as Bogomolov multiplier and we are going to use his terminology here. Denote by b ( G ) the order of the group B ( G ) and denote by h ( G ) the Schur multiplier of the group G , that is, the order ofthe group H ( G, Z ).The aim of this article is to prove Theorem 1.
For an equipped finite group ( G, O ) we have the following inequalities b ( G ) a ( G,O ) h ( G ) . In particular, a ( G,G \{ } ) = b ( G ) . Since, by [8], b ( G ) = 1 for a finite almost simple group G , we conclude: Corollary 1.
Let G be a finite almost simple group. Then there is a constant T such that for any projective irreducible non-singular curve F each non-empty Hurwitzspace HUR md,G,G \{ } ,τ ( F, z ) is irreducible if all τ i > T . HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 3
It was shown in [10] that if O ⊂ O are two equipments of a finite group G , then a ( G,O ) a ( G,O ) .For a symmetric group Σ d , the famous Clebsch – Hurwitz Theorem ([3], [6]) statesthat the ambiguity index a (Σ d ,T ) = 1, where T is the set of transpositions in Σ d ,and it was shown in [11] that the ambiguity index a (Σ d ,O ) = 1 if the equipment O contains an odd permutation σ ∈ Σ d such that σ leaves fixed at least two elements.Theorem 8 (see subsection 3.4) gives the complete answer on the value of a (Σ d ,O ) foreach equipment O of Σ d . Also in subsection 3.4, we give the complete answer on thevalue of a ( A d ,O ) for each d and for each equipment O of the alternating group A d .In Section 1, we remind some properties of C -groups and prove one of the inequal-ities claimed in Theorem 1. In Section 2, we complete the proof of this Theorem.In Section 3, we investigate the properties of ambiguity indices of a quasi-cover ofan equipped finite group ( G, O ) and in Section 4, we give a cohomologycal descriptionof the ambiguity indices.In Section 5, we give examples of finite groups G which Bogomolov multiplier b ( G ) >
1. Therefore for such groups G each non-empty space HUR md,G,O,τ ( F, z )consists of at least b ( G ) > τ = ( τ , . . . , τ m ) withbig enough τ i .In this article, if F is a free group freely generated by an alphabet X , N is a normalsubgroup of F , and a group G = F /N , then a word w = w ( x i , . . . , x i n ) in letters x i j ∈ X and their inverses will be considered as an element of G in case if it does notlead to misunderstanding.1. C -groups and their properties Let us remind the definition of a C -group (see, for example, [12]). Definition 2.
A group G is a C -group if there is a set of generators x ∈ X in G such that the basis of relations between x ∈ X consists of the following relations: x − i x j x i = x k , ( x i , x j , x k ) ∈ M, (2) where M is a subset of X . Thus the C -structure of G is defined by X ⊂ G and M ⊂ X .Let F be a free group freely generated by an alphabet X . Denote by N the subgroupof F normally generated by the elements x − i x j x i x − k , ( x i , x j , x k ) ∈ M . The group N is a normal subgroup of F . Let f : F → G = F /N be the natural epimorphism givenby presentation (2). In the sequel, we consider each C -group G as an equipped group( G, O ) with the equipment O = f ( X F ) (where X F is the orbit of X under the actionof the group of inner automorphisms of F ). The elements of O are called C - generators of the C -group G . In particular, the equipped group ( F , X F ) is a C -group.A homomorphism f : G → G of a C -group ( G , O ) to a C -group ( G , O )is called a C - homomorphism if it is a homomorphism of equipped groups, that is, f ( O ) ⊂ O . In particular, two C -groups ( G , O ) and ( G , O ) are C - isomorphic ifthey are isomorphic as equipped groups. F.A. BOGOMOLOV AND VIK.S. KULIKOV
Claim 1. ( Lemma 3.6 in [12] ) Let N be a normal subgroup of F normally generatedby elements of the form w − i x j w i w l x − k w − l , where w i and w l are some elements of F and x j , x k ∈ X . Let f : F → G ≃ F /N be the natural epimorphism. Then ( G, f ( X F )) is a C -group and f is a C -homomorphism. To each C -group ( G, O ), one can associate a C -graph. By definition, the C - graph Γ = Γ ( G,O ) of a C -group ( G, O ) is a directed labeled graph whose set of vertices V = { v g i | g i ∈ O } is in one to one correspondence with the set O . Two vertices v g and v g , g , g ∈ O , are connected by a labeled edge e v g ,v g ,v g (here v g is the tail of e v g ,v g ,v g , v g is the head of e v g ,v g ,v g , and v g is the label of e v g ,v g ,v g ) if and only if in G we have the relation g − g g = g with some g ∈ O .A C -homomorphism f : ( G , O ) → ( G , O ) of C -groups induces a map f ∗ :Γ ( G ,O ) → Γ ( G ,O ) from the C -graph Γ ( G ,O ) in the C -graph Γ ( G ,O ) , where by defi-nition, f ∗ ( v g ) = v f ( g ) for each vertex v g of Γ ( G ,O ) and f ∗ ( e v g ,v g ,v g ) = e v f ( g ,v f ( g ,v f ( g ) for each edge e v g ,v g ,v g of Γ ( G ,O ) .The following Claim is obvious. Claim 2. A C -homomorphism f : ( G , O ) → ( G , O ) is a C -isomorphism if f ∗ isone-to-one between the sets of vertices of Γ ( G ,O ) and Γ ( G ,O ) . In the sequel, we will consider only finitely presented C -groups (as groups withoutequipment) and C -graphs consisting of finitely many connected components. Denoteby m the number of connected components of a C -graph Γ ( G,O ) .Then it is easy to see that G/ [ G, G ] ≃ Z m and any two C -generators g and g areconjugated in the C -group G if and only if v g and v g belong to the same connectedcomponent of Γ ( G,O ) .Thus the set O of C -generators of the C -group ( G, O ) is the union of m conjugacyclasses of G and there is a one-to-one correspondence between the conjugacy classesof G contained in O and the set of connected components of Γ ( G,O ) .Denote by τ : G Γ → H ( G, Z ) = G/ [ G, G ] the natural epimorphism. In the se-quel, we fix some numbering of the connected components of Γ ( G,O ) . Then the group H ( G, Z ) ≃ Z m obtains a natural base consisting of vectors τ ( g ) = (0 , . . . , , , . . . , i -th place if g is a C -generator of G and v g belongs to the i -thconnected component of Γ ( G,O ) . For g ∈ G the image τ ( g ) is called the type of g . Lemma 1.
Let g , g be two C -generators of a C -group ( G, O ) , N the normal closureof g g − in G , and f : G → G = G/N the natural epimorphism. Then ( i ) ( G , O ) is a C -group, where O = f ( O ) , and f is a C -homomorphism; ( ii ) the map f ∗ : Γ ( G,O ) → Γ ( G ,O ) is a surjection. ( iii ) if g g − belong to the center Z ( G ) of the group G and let v g and v g belongto different components of Γ ( G,O ) , then ( iii ) the number of connected components of the C -graph Γ ( G ,O ) is less thanthe number of connected components of the C -graph Γ ( G,O ) ; ( iii ) f : [ G, G ] → [ G , G ] is an isomorphism. HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 5
Proof.
Claims ( i ), ( ii ), and ( iii ) are obvious.To prove ( iii ), note that N is a cyclic group generated by g g − , since g g − be-longs to the center Z ( G ). The type τ (( g g − ) n ) is non-zero for n = 0, since v g and v g belong to different connected components of Γ ( G,O ) . Therefore to complete theproof, it suffices to note that the groups N and [ G, G ] have trivial intersection, since τ ( g ) = 0 for all g ∈ [ G, G ]. (cid:3) A C -group ( G, O ) is called a C - finite group if the set of vertices of C -graph Γ ( G,O ) is finite or, the same, if the equipment O of G is a finite set. Proposition 1. ([10])
Let ( G, O ) be a C -finite group. Then the commutator [ G, G ] is a finite group. As it was mention in Introduction, for each finite equipped group (
G, O ), one canassociate a C -group ( e G, e O ) defined as follows. The group e G is generated by the lettersof the alphabet Y = Y O = { y g | g ∈ O } subject to relations y g y g = y g y g − g g = y g g g − y g . (3)Here e O = Y O and there is a natural epimorphism β O : e G → G given by β O ( y g ) = g .Note also that a homomorphism of equipped groups f : ( G , O ) → ( G, O ) inducesa C -homomorphism e f : ( e G , e O ) → ( e G, e O ) such that f ◦ β O = e f ◦ β O .Let the elements of a subset S of an equipment O of a group G generate thegroup G and O = S G . Denote by F S a free group freely generated by the alphabet Y S = { y g | g ∈ S } and R S is the normal subgroup of F S such that the naturalepimorphism h S : F S → F S /R S ≃ G gives a presentation of the group G . Claim 3.
Let e R S ⊂ R S be the normal subgroup normally generated by the elementsof R S of the form w − i,j y g i w i,j y − g j , where w i,j ∈ F S and y g i , y g j ∈ Y S . Then the C -group ( e G, e O ) has the following presentation: e G ≃ F S / e R S such that the images of theelements of Y S are C -generators of e G .Proof. Denote by G = F S / e R S . By Claim 1, G is a C -group with C -equipment O = Y G S and there is a natural epimorphism β S : (( G , O ) → ( G, O ) given by β S ( y g ) = g for g ∈ S .Assume that S consists of elements g , . . . , g n ∈ O . If S = O then choose anelement g n +1 ∈ O \ S . It is conjugated to some g i ∈ S . Denote by R g n +1 the set of allpresentations of g n +1 in the form g n +1 = w ( g , . . . , g n ) − gw ( g , . . . , g n ) , g ∈ S. (4)Note that if g n +1 = w i ( g , . . . , g n ) − g i w i ( g , . . . , g n ) and g n +1 = w j ( g , . . . , g n ) − g j w j ( g , . . . , g n ) , then w j w − i g i w i w − j = g j , F.A. BOGOMOLOV AND VIK.S. KULIKOV that is, w j ( y g , . . . , y g n ) w − i ( y g , . . . , y g n ) y g i w i ( y g , . . . , y g n ) w − j ( y g , . . . , y g n ) y − g j ∈ R S . (5)Similarly, if g n +1 = w i ( g , . . . , g n ) and g − n +1 g i g n +1 = g j for some g i , g j ∈ S , then w ( y g , . . . , y g n ) − y g i w ( y g , . . . , y g n ) y − g j ∈ R S . (6)Therefore, if S = S ∪ { g n +1 } , F S is a free group freely generated by the alphabet Y S = { y g | g ∈ S } , R g n +1 is the set of words of the form w ( y g , . . . , yg n ) − y g w ( y g , . . . , y g n ) y − g n +1 defined by all relations (4), and e R S is the normal closure in F S of the set e R S ∪ Rg n +1 ,then G ≃ F S / e R S in view of relations (5) and (6).Note that if we have a relation g − i g j g i = g k for some g i , g j , g k ∈ S then y − g i y g j y g i y − g k ∈ e R S . (7)If S = O , then we can repeat the construction described above and obtain a pre-sentation G ≃ F S / e R S , and so on. After several steps we obtain a presentation G ≃ F O / e R O . Note that, by induction, we deduce that for any relation in G of theform g − i g j g i = g k for some g i , g j , g k ∈ O we have y − g i y g j y g i y − g k ∈ e R O . Thereforethere is a natural C -homomorphism f : ( e G, e O ) → ( G , O ). By Claim 2, f is a C -isomorphism. (cid:3) For an equipped finite group (
C, O ), consider a presentation of G of the followingform. Let us take a free group F = F O freely generated by the alphabet X O = { x g | g ∈ O } . Consider a normal subgroup R O ⊂ F such that F /R O ≃ G . Let h O : F → F /R O ≃ G be the natural epimorphism.We can associate to ( G, O ) a group G = F / [ F , R O ].Denote by α O : G → G the natural epimorphism. By Claim 1, ( G, O ) is a C - group,where O = h O ( X F O ). It is evident that there is the natural epimorphism of C -groups κ O : ( G, O ) → ( e G, e O ) sending κ O ( x g ) = y g for all g ∈ O and such that α O = β O ◦ κ O .The C -group ( G, O ) is called the universal central C -extension of the equipped finitegroup ( G, O ).It is easy to see that α O : G → G is a central extension of groups, that is, ker α O is a subgroup of the center Z ( G ).We have ker α O ∩ [ G, G ] = ( R O ∩ [ F , F ]) / [ F , R O ] . By Hopf’s integral homology formula, we have H ( G, Z ) ≃ ( R O ∩ [ F , F ]) / [ F , R O ].Denote by h ( G ) the order of the group H ( G, Z ) and denote by K ( G,O ) the subgroupof ( R O ∩ [ F , F ]) / [ F , R O ] generated by the elements of R O of the form [ w, x g ], where g ∈ O and w ∈ F , and let k ( G,O ) be its order. HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 7
Theorem 3.
For an equipped finite group ( G, O ) we have h ( G ) = k ( G,O ) a ( G,O ) . Proof.
We have ker κ O ⊂ ker α O . Therefore ker κ O ⊂ Z ( G ).Let us show that for some n > C -groups G = F /R , . . . , G n = F /R n , a sequence of C -homomorphisms ϕ i : ( G i , O i ) → ( G i +1 , O i +1 ) , i n − , where ( G , O ) = ( G, O ), and a C -homomorphism κ : ( G n , O n ) → ( e G, e O ) such that( i ) κ = κ ◦ ϕ , where ϕ = ϕ n − ◦ · · · ◦ ϕ ;( ii ) for each i the homomorphism ϕ i : [ G i , G i ] → [ G i +1 , G i +1 ] is an isomorphism;( iii ) κ ∗ induces a one-to-one correspondence between the connected components ofthe C -graphs Γ ( G n ,O n ) and Γ ( e G, e O ) .Indeed, let us put R = R O and consider the map κ ∗ . If it is induces a one-to-onecorrespondence between the connected components of the C -graphs Γ ( G,O ) and Γ ( e G, e O ) ,then n = 0 and it is nothing to prove.Otherwise, for some g ∈ O there is a vertex v y g of Γ ( e G, e O ) which preimage κ − ∗ ( v y g )contains at least two vertices, say v x g and v g (here g is an element of X F ), of Γ ( G,O ) belonging to different connected components of Γ ( G,O ) .Denote by R the normal closure of R O ∪ { x g g − } in F and consider the naturalhomomorphism ϕ : G → G = F /R . The element x g g − , considered as an elementof G , belongs to ker κ . Therefore, x g g − ∈ Z ( G ).Denote by κ : G → e G the homomorphism induced by κ . By Lemma 1, thehomomorphism ϕ is a C -homomorphism of C -groups. It is easy to see that ϕ :[ G , G ] → [ G , G ] is an isomorphism and the number of connected components ofthe C -graph Γ ( G ,O ) is less than the number of connected components of the C -graphΓ ( G,O ) .Assume now that κ ∗ is not a one-to-one correspondence between the connectedcomponents of the C -graphs Γ ( G ,O and Γ ( e G, e O ) . Then for some g ∈ O there is avertex v y g of Γ ( e G, e O ) which preimage κ − ∗ ( v y g ) contains at least two vertices v x g and v g of Γ ( G ,O ) belonging to different connected components of Γ ( G ,O ) .Hence we can repeat the construction described above and obtain a C -group( G , O ) and C -homomorphisms ϕ : G → G = F /R and κ : G → e G suchthat ϕ : [ G , G ] → [ G , G ] is an isomorphism and the number of connected compo-nents of the C -graph Γ ( G ,O ) is less than the number of connected components of the C -graph Γ ( G ,O ) . Since the number of connected components of the C -graph Γ ( G,O ) is finite, after several ( n ) steps of our construction we obtain the desired sequences of C -groups and C -homomorphisms.Now, consider the C -homomorphism κ : G n → e G . The C -graph Γ ( e G, e O ) consists ofconnected components Γ , . . . , Γ m . Let { v g i, , . . . , v g i,li } be the set of the vertices of F.A. BOGOMOLOV AND VIK.S. KULIKOV Γ i . We have O = { g i,j } i m, j l i . Then Γ i = κ − ∗ (Γ i ) are the connected componentsof Γ ( G n ,O n ) .Let κ − n ( v y gi,j ) = { v x gi,j , v g i,j, , . . . , v g i,j,ri,j } , g i,j,k ∈ O n for 1 k r i,j .Since the graph Γ i is connected, there are words w i,j,k in letters of X O and theirinverses such that g i,j,k = w i,j,k x g i,j w − i,j,k , k r i,j . Obviously, the elements u i,j,k = [ w i,j,k , x g i,j ] = g i,j,k x − g i,j belong to [ G n , G n ] ∩ ker κ .Therefore u i,j,k , as elements of F belong to R O ∩ [ F , F ].Consider the group G n +1 = F /R n +1 , where the group R n +1 is the normal closureof R n ∪ { u i,j,k } i m, j l i , k r i,j in F . Then, by Claim 1, G n +1 = F /R n +1 is a C -group and the natural map κ : G n +1 → e G , induced by κ , is a C -homomorphism.Moreover, ker ϕ n of the natural epimorphism ϕ n : G n → G n +1 is a subgroup of[ G n , G n ] ≃ [ G, G ] = [ F , F ] / [ F , R O ] generated by the elements u i,j,k = [ w i,j,k , x g i,j ],where 1 i m , 1 j l i , and 1 k r i,j .To complete the proof of Theorem 3, it suffices to note that κ ∗ induces a one-to-onecorrespondence between the sets of vertices of the C -graphs Γ ( G n +1 ,O n +1 ) and Γ ( e G, e O ) ,since all u i,j,k = g i,j,k x − g i,j belong to ker ϕ n . Therefore κ is an isomorphism. (cid:3) Lemma 2.
Let the order of g ∈ O be n and let [ x g , w ] ∈ ([ F , F ] ∩ R O ) / [ F , R O ] ⊂ F / [ F , R O ] . Then the order of the element [ x g , w ] is a divisor of n .Proof. The elements x ng and [ x g , w ] belong to the center of the group F / [ F , R O ]. There-fore [ x ng , w ] = x n − g [ x g , w ] x − ng [ x n − g , w ] = [ x g , w ][ x n − g , w ] = · · · = [ x g , w ] n is the unity of F / [ F , R O ]. (cid:3) Proposition 2.
Let the equipment O of an equipped finite group ( G, O ) consists ofconjugacy classes of elements of orders coprime with h ( G ) . Then a ( G,O ) = h ( G ) .Proof. It follows from Lemma 2 and Theorem 3. (cid:3) Proof of Theorem 1
By definition, the
Bogomolov multiplier b ( G ) of a finite group G is the order ofthe group B ( G ) = ker[ H ( G, Q / Z ) → O A ⊂ G H ( A, Q / Z )]where A runs over all abelian subgroups of G . Remark 1.
Note that it suffices to consider only restrictions to abelian groups withtwo generators in order to define that the element w ∈ H ( G, Q / Z ) is contained in B ( G ). HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 9
There is a natural duality between H ( G, Q / Z ) and H ( G, Z ). Both groups arefinite for finite groups G and duality implies an isomorphism of H ( G, Q / Z ) and Hom ( H ( G, Z ) , Q / Z ) as abstract groups.By Theorem 3, we have the inequality h ( G ) > a ( G,O ) for any equipped finitegroup ( G, O ). By Corollary 2 in [10], we have inequality a ( G,O ) > a ( G,G \{ } ) for eachequipment O of G . Therefore to prove Theorem 1 it suffice to show that for theequipped finite group ( G, G \ { } ) its ambiguity index a ( G,G \{ } ) is equal to b ( G ).In notation used in Section 1 and by Theorem 3, we have a ( G,G \{ } ) = h ( G ) k ( G,G \{ } ) , where k ( G,G \{ } ) is the order of the subgroup K G \{ } of the group( R G \{ } ∩ [ F G \{ } , F G \{ } ]) / [ F G \{ } , R G \{ } ] ≃ H ( G, Z )generated by the elements of R G \{ } of the form [ w, x g ], where g ∈ G \ { } and w ∈ F G \{ } . Lemma 3.
Let for some w , w ∈ F G \{ } the commutator [ w , w ] belong to R G \{ } .Then [ w , w ] , considered as an element of F G \{ } / [ F G \{ } , R G \{ } ] , belongs to K G \{ } .Proof. First of all, note that if [ x g , w ] ∈ K G \{ } , then [ x g , w ] = [ w, x − g ] = [ x − g , w − ] =[ x − g , w ] in K G \{ } , since K G \{ } is a subgroup of the center of the C -group G G \{ } = F G \{ } / [ F G \{ } , R G \{ } ] and these four commutators are conjugated to each other in F G \{ } . Similarly, [ w, x g ] = [ x g , w − ] = [ w − , x − g ] = [ x − g , w − ] ∈ K G \{ } , since [ w, x g ]is the inverse element to the element [ x g , w ]. Note also that for any w the element w [ w, x g ] w − belongs to K G \{ } if [ w, x g ] ∈ K G \{ } .Next, the elements w − and w − , considered as elements of G , are equal to someelements g and g of G . Therefore if [ w , w ] ∈ R G \{ } then w x g , w x g , [ x g , x g ] , [ w , x g ] , [ w , x g ] ∈ R G \{ } . In addition, we have [ w , w x g ] ∈ [ F G \{ } , R G \{ } ] and[ w , w x g ] = [ w , w ]( w [ w , x g ] w − ) . Therefore [ w , w ] ∈ R G \{ } ∩ [ F G \{ } , F G \{ } ] (as an element of K G \{ } ) is the inverseelement to the element [ w , x g ] ∈ K G \{ } and hence [ w , w ] ∈ K G \{ } . (cid:3) To complete the proof of Theorem 1, note that, by Lemma 3, for each imbedding i : H → G of an abelian group H generated by two elements the image of i ∗ : H ( H, Z ) → H ( G, Z ) is a subgroup of K G \{ } and the group K G \{ } is generated bythe images of such elements. Therefore the group K ⊥ G \{ } = { ϕ ∈ Hom ( H ( G, Z ) , Q / Z ) | ϕ ( w ) = 0 for all w ∈ K G \{ } } coincides with the group B ( G ) and its order is equal to a ( G,G \{ } ) = h ( G ) k ( G,G \{ } ) . (cid:3) Quasi-covers of equipped finite groups
In this section we use notations introduced in Sections 1.3.1.
Definitions.
Let f : ( G , O ) → ( G, O ) be a homomorphism of equipped groups.We say that f is a cover of equipped groups (or, equivalently, ( G , O ) is a cover of( G, O )) if( i ) f is an epimorphism such that f ( O ) = O ;( ii ) ker f is a subgroup of the center ZG of G ;( iii ) f ∗ : H ( G , Z ) → H ( G, Z ) is an isomorphism.Let f : ( G , O ) → ( G, O ) be a homomorphism of equipped finite groups. We saythat S ⊂ O is a section of f if f | S : S → O is a one-to-one correspondence. Denoteby O S ⊂ O the orbit of S under the action of the group of the inner automorphismsof G .Let f : ( G , O ) → ( G, O ) be an epimorphism of equipped groups such that ker f ⊂ ZG . We say that f is a quasi-cover of equipped groups (or, equivalently, ( G , O ) isa quasi-cover of ( G, O )) if there is a section S of f such that O S = O .Below, we will assume that for a quasi-cover f of equipped groups a section S ischosen and fixed.3.2. Properties of quasi-covers.Lemma 4.
Let f : ( G , O ) → ( G, O ) be a cover of equipped finite groups and S ⊂ O a section. Then G is generated by the elements of S .Proof. Denote by G S the subgroup of G generated by the elements of S . Obviously, ϕ = f | G S : G S → G is an epimorphism and ker ϕ ⊂ ker f . Therefore, to prove Lemmait suffices to show that ker f ⊂ G S . To show this, let us consider the natural epimor-phism f : G → G = G / ker ϕ and the natural epimorphism ψ : G → G inducedby f . Obviously, ψ : ( G , f ( O )) → ( G, O ) is a cover of equipped finite groups and ψ | H : H → G is an isomorphism, where H = f ( G S ). Therefore G ≃ ker ψ × G .Consequently, ker ψ = 0, since ψ ∗ : H ( G , Z ) → H ( G, Z ) is an isomorphism. (cid:3) If S is a section of a cover f : ( G , O ) → ( G, O ), then Lemma 4 implies that O S = S G is an equipment of G and f : ( G , O S ) → ( G, O ) is also a cover ofequipped groups.Below, we fix a section S of a cover f : ( G , O ) → ( G, O ). Then the cover f canbe considered as a quasi-cover.In notations used in Section 1, consider the universal central C -extension α O :( G, O ) → ( G, O ) of an equipped finite group (
G, O ). We have two natural epi-morphisms h O : F O → G = F O /R O and β O : F O → G = F O / [ F O , R O ] such that h O = α O ◦ β O . HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 11
Lemma 5.
Let f : ( G , O ) → ( G, O ) be a quasi-cover of equipped finite groups.Then there is an epimorphism α S : ( G, O ) → ( G , O S ) of equipped groups such that α O = f ◦ α S .Proof. By Lemma 4, there is an epimorphism h S : F O → G defined by h S ( x g ) = b g ∈ S for all g ∈ G , where b g = f − | S ( g ). Denote by R S = ker h S . Obviously, we have f ◦ h S = h O . Therefore R S ⊂ R O .Let us show that the group [ F O , R O ] is a subgroup of R S . Indeed, consider any w ∈ R O . Then, as an element of G , the element w ∈ ker f and, consequently, w belongs to the center of G . In particular, it commutes with any generator b g ∈ S of G and hence [ w, x g ] ∈ R S , that is, [ F O , R O ] ⊂ R S .The inclusion [ F O , R O ] ⊂ R S implies the desired epimorphism α S . (cid:3) We say that a cover (resp., a quasi-cover) of equipped finite groups f : ( G , O ) → ( G, O ) is maximal if for any cover of equipped finite groups f : ( G , O ) → ( G , O )such that f = f ◦ f is also a cover (resp., quasi-cover) of equipped finite groups, theepimorphism f is an isomorphism. Theorem 4.
For any cover ( resp., quasi-cover ) of equipped finite groups f : ( G , O ) → ( G, O ) , there is a maximal cover ( resp., quasi cover ) f : ( G , O ) → ( G, O ) for whichthere is a cover f : ( G , O ) → ( G , O S ) such that ( i ) f = f ◦ f ; ( ii ) ker f ≃ H ( G, Z ) ( resp., [ G, G ] ∩ ker f ≃ H ( G, Z )) .Proof. Consider the epimorphism α S : ( G, O ) → ( G , O S ) defined in the proof ofLemma 5. The group ker α S is a subgroup of the center of G .Since ( G, O ) is a C -group and O consists of M conjugacy classes, where M | O | = rk F O , then H ( G, Z ) = G/ [ G, G ] = Z M . Let τ : G → Z M be the natural ho-momorphism (that is, τ is the type homomorphism G → H ( G, Z ), see Introduction).The image τ (ker α S ) is a sublattice of maximal rank in Z M . Let us choose a Z -freebasis a , . . . , a M in τ (ker α S ) and choose elements g i ∈ ker α S , 1 i M , such that τ ( g i ) = a i .Denote by H S a group generated by the elements g i , 1 i M , and denote by K S = [ G, G ] ∩ ker α S . Then it is easy to see that H S ≃ Z M is a subgroup of the centerof G , the intersection H S ∩ [ G, G ] is trivial, and ker α S ≃ K S × H S .Denote by G = G/H S the quotient group and by α H S : G → G , f : G → G the natural epimorphisms. We have α S = f ◦ α H S . Denote also by O = α H S ( O ).Then it is easy to see that α H S : ( G, O ) → ( G , O ) and f : ( G , O ) → ( G , O S ) arecentral extensions of equipped groups.By construction, it is easy to see that [ G, G ] ∩ ker α H S is trivial and ker f ⊂ [ G , G ]is a subgroup of the center of G . Therefore the epimorphism f is a cover of equipped groups. In addition, it is easy to see that α O = f ◦ α H S and f = f ◦ f : ( G , O ) → ( G, O ) is a cover (resp., quasi-cover) of equipped groups. We have K S ≃ ker f ⊂ α H O ([ G, G ] ∩ ker α ) = α H O ( H ( G, Z )) ⊂ [ G , G ] . Therefore, if k f i = | ker f i | , i = 1 ,
2, is the order of the group ker f i and k f is the orderof ker f , then h ( G ) = k f = k f k f . (8)Since we can repeat the construction described above to the cover (resp., quasi-cover) f and applying again equality (8), where new f is our f and new f is a coverexistence of which follows from assumption that old f is not maximal, we obtainthat new f is an isomorphism, that is, the covering f is maximal. (cid:3) In the case then f : ( G, G \ { } ) → ( G, G \ { } ) is an isomorphism of equippedfinite groups, a maximal cover f : ( G , O ) → ( G, G \ { } ), constructed in the proofof Theorem 4, will be called a universal maximal cover . Corollary 2.
For any equipped finite group ( G, O ) there is a maximal cover ofequipped groups.For any cover ( resp., quasi-cover ) f : ( G , O ) → ( G, O ) of equipped finite groups, k f = | ker f | h ( G ) (resp., k f = | ker f ∩ [ G , G ] | h ( G ) ) and f is maximal ifand only if k f = h ( G ) . The ambiguity index of a quasi-cover of equipped group.
Let ( e G, e O ) bethe C -group associated with an equipped group ( G, O ) and β O : ( e G, e O ) → ( G, O ) thenatural epimorphism of equipped groups (see definitions in Section 1).
Theorem 5.
Let f : ( G , O ) → ( G, O ) be a quasi-cover of equipped finite groups.Then there is a natural C -epimorphism κ S : ( G, O ) → ( e G , e O S ) such that κ O = e f ◦ κ S and α O = β O ◦ e f ◦ κ S = f ◦ β O S ◦ κ S , where the C -epimorphism κ O : ( G, O ) → ( e G, e O ) is defined in Section 1 and the C -epimorphism e f : ( e G , e O S ) → ( e G, e O ) is associatedwith f .Proof. In notations used in the proof of Lemma 5, we have an inclusion R S ⊂ R O ofnormal subgroups of F O which induces f : G = F O /R S → G = F O /R O .Let e R S ⊂ R S be the normal subgroup normally generated by the elements of R S of the form w − i,j x g i w i,j x − g j , where w i,j ∈ F O and x g i , x g j ∈ X O . For any w ∈ R O andany generator x g , g ∈ O , the commutator [ x g , w ] ∈ R S , since f is a central extensionof groups. Therefore [ F O , R O ] ⊂ e R S . (9)By Claim 3, e G ≃ F S / e R S . Therefore inclusion (9) induces an epimorphism κ S : G = F O / [ F O , R O ] → F / e R S ≃ e G Obviously, the C -epimorphism κ S : ( G, O ) → ( e G , e O S )satisfies all properties claimed in Theorem 5. (cid:3) HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 13
Let f : ( G , O ) → ( G, O ) be a cover (resp., quasi-cover) of equipped finite groupsand e f S : ( e G , e O S ) → ( e G, e O ) C -epimorphism associated with f : ( G , O S ) → ( G, O ).Denote by k f the order of the group ker f ∩ [ G , G ] and by k e f S the order of the groupker e f S ∩ [ e G , e G ]. Corollary 3.
Let f : ( G , O ) → ( G, O ) be a quasi-cover of equipped finite groups, S a section of f . Then h ( G ) = a ( G,O ) k e f S k S = k f a ( G ,O S ) k S , where k S is the order of the group ker κ S ∩ [ G, G ] . Corollary 4.
Let f : ( G , O ) → ( G, O ) be a cover ( resp., quasi-cover ) of equippedfinite groups, S a section of f . Then for any equipment b O of G ( resp., such that O ⊂ b O ) we have an inequality a ( G , b O ) h ( G ) .If f is maximal, then a ( G , b O ) = 1 .Proof. If f is a cover, then f : ( G , b O ) → ( G, f ( b O )) is also a cover of equipped groupsand a ( G , b O ) h ( G ) by Corollary 3.As it was mention in the Introduction, we have a ( G , b O ) a ( G ,O ) if O ⊂ b O and if f is a quasi-cover, then a ( G ,O ) h ( G ) by Corollary 3.If f is maximal, then k f = h ( G ) by Corollary 2 and therefore if f is a cover then f : ( G , b O ) → ( G, f ( b O )) is also maximal. It follows from Corollary 3 that a ( G , b O ) = 1in the case of maximal covers, and a ( G , b O ) a ( G ,O ) = 1 in the case of maximalquasi-covers f . (cid:3) Let f : ( G , O ) → ( G, O ) be a cover of equipped finite groups such that f − ( O ) = O . We say that f splits over a conjugacy class C ⊂ O if f − ( C ) consists of at leasttwo conjugacy classes of G . The number s f ( C ) of the conjugacy classes containingin f − ( C ) is called the splitting number of the conjugacy class C for f . We say that f splits completely over C if s f ( C ) = k f , where k f = | ker f | .Let C be a conjugacy class in G . Consider the subgroups K C ⊂ K G \{ } of thegroup ( R G \{ } ∩ [ F G \{ } , F G \{ } ]) / [ F G \{ } , R G \{ } ] ≃ H ( G, Z ) , where K C is generated by the elements of R G \{ } of the form [ x h , x g ], h ∈ G \ { } .Let k C be the order of the group K C . Proposition 3.
Let f : ( G , O ) → ( G, G \ { } ) be a universal maximal cover ofequipped finite groups and let C be a conjugacy class in G . Then h ( G ) = s f ( C ) k C .Proof. For g ∈ C the preimage f − ( C ) consists of the conjugacy classes of the elements zx g , where z ∈ ker f = ( R G \{ } ∩ [ F G \{ } , F G \{ } ]) / [ F G \{ } , R G \{ } ] ≃ H ( G, Z ). Notethat ker f ⊂ ZG and ker f acts transitively on the set of the conjugacy classes C , . . . C k f ( C ) involving in f − ( C ), z ( C i ) = C j if zg ∈ C j for g ∈ C i . Let x g ∈ C , where g ∈ C . Then z ( C ) = C if and only if for some w ∈ G wehave wx g w − = zx g , that is, z = [ w, x g ].If f ( w ) = h then w = z x h for some z ∈ ker f and therefore z = [ x h , x g ], that is, z ∈ K C . The inverse statement that each element z ∈ K C leaves fixed the conjugacyclass C is obvious. (cid:3) Proposition 4.
Let f : ( G , O ) → ( G, G \ { } ) be a universal maximal cover ofequipped finite groups. Then a ( G,O ) = h ( G ) if and only if f splits completely overeach conjugacy class C ⊂ O .If s f ( C ) = 1 for some conjugacy class C ⊂ O then a ( G,O ) = 1 .Proof. We have k f = h ( G ).The map g x g is a section in O . Denote by O the equipment of G consisting ofthe elements conjugated to x g , g ∈ O . Therefore f : ( G , O ) → ( G, O ) is a maximalcover of equipped groups and Proposition 4 follows from Corollary 3. (cid:3)
Proposition 5.
Let f : ( G , O ) → ( G, G \ { } ) be a universal maximal cover ofequipped finite groups and let C ⊂ O and C ⊂ O be two conjugacy classes containingin an equipment of G . Then a ( G,O ) = 1 if s f ( C ) and s f ( C ) are coprime.Proof. Follows from Corollary 3, since the group ker e f S ∩ [ e G , e G ] ⊂ H ( G, Z ) containstwo subgroups K C and K C whose indices in H ( G, Z ) are coprime. (cid:3) Proposition 6.
Let f : ( G , O ) → ( G, G \ { } ) be a universal maximal cover ofequipped finite groups and let h ( G ) = pq , where p and q are coprime integers. Let C ⊂ O be a conjugacy class such that s f ( C ) = q and let s f ( C ) is coprime with p foreach conjugacy class C ⊂ O . Then the ambiguity index a ( G,O ) = p .Proof. Follows from Corollary 3, since the group ker e f S ∩ [ e G , e G ] ⊂ H ( G, Z ) gener-ated by the subgroups K C of index p in ker f and subgroups of indices also coprimewith p . (cid:3) The ambiguity indices of symmetric groups and alternating groups.
In[5], it was proved the following theorems
Theorem 6. (Theorem 3.8 in [5])
Let e Σ d be a maximal cover of the symmetric group Σ d . The conjugacy classes of Σ d which split in e Σ d are: (a) the classes of even per-mutations which can be written as a product of disjoint cycles with no cycles of evenlength; and (b) the classes of odd permutations which can be written as a product ofdisjoint cycles with no two cycles of the same length ( including 1 ) . Theorem 7. (Theorem 3.9 in [5])
Let e A d be the maximal cover of the alternatinggroup A d . The conjugacy classes of A d which split in e A d are: (a) the classes ofpermutations whose decompositions into disjoint cycles have no cycles of even length;and (b) the classes of permutations which can be expressed as a product of disjointcycles with at least one cycle of even length and with no two cycles of the same length ( including 1 ) . HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 15
Remind that, by definition, an equipment O of Σ d must contain a conjugacy classof odd permutation since the elements of the equipment must generate the group.It is well known that for the symmetric group Σ d , d >
4, and for the alternatinggroup A d , d = 6 , d >
4, the Schur multiplier h (Σ d ) = h ( A d ) = 2. The followingtheorems are straightforward consequences of Proposition 3 and Theorems 4 – 7. Theorem 8.
Let O be an equipment of a symmetric group Σ d . Then a (Σ d ,O ) = 2 if and only if O consists of conjugacy classes of odd permutations such that theycan be written as a product of disjoint cycles with no two cycles of the same length ( including 1 ) and conjugacy classes of even permutations such that they can be writtenas a product of disjoint cycles with no cycles of even length. Overwise, a (Σ d ,O ) = 1 . Theorem 9.
Let O be an equipment of an alternating group A d , d = 6 , . Then a ( A d ,O ) = 2 if and only if O consists of conjugacy classes of permutations whosedecompositions into disjoint cycles have no cycles of even length and the classes ofpermutations which can be expressed as a product of disjoint cycles with at least onecycle of even length and with no two cycles of the same length ( including 1 ) . Overwise, a ( A d ,O ) = 1 . It is well known that in the case when d = 6 ,
7, the Schur multiplier h ( A d ) = 6.For σ ∈ A d denote by c ( σ ) = ( l , . . . , l m ) the cycle type of permutation σ , thatis, the collection of lengths l i of non-trivial (that is l i >
2) cycles entering into thefactorization of σ as a product of disjoint cycles. For a conjugacy class C in A d thecollection c ( C ) = c ( σ ) is called the cycle type of C if σ ∈ C . It is well known that thecycle type c ( C ) does not depend on the choice of σ ∈ C and there are at most twoconjugacy classes in A d of a given cycle type c .The group A d , d = 6 ,
7, has the following non-trivial conjugacy classes:(I) two conjugacy classes of each cycle type (5), (2 , d = 7) (7);(II) two conjugacy classes of cycle type (3) and one conjugacy class of cycle type(3 , ,
2) and one conjugacy class of cycle type(2 , ,
3) if d = 7. Proposition 7.
The ambiguity index a ( A d ,O ) , d = 6 , , takes the following values: (I) a ( A d ,O ) = 6 if O contains only the elements of conjugacy classes of type (I) ; (II) a ( A d ,O ) = 2 if O contains only the elements of conjugacy classes of type (I) andthe elements of at least one conjugacy class of type (II) ; (III) a ( A d ,O ) = 3 if O contains only the elements of conjugacy classes of type (I) andthe elements of at least one conjugacy class of type (III) ; (II+III) a ( A d ,O ) = 1 if O contains the elements of at least one conjugacy class of type (II) and the elements of at least one conjugacy class of type (III) . Proof.
Let f : ( G , O ) → ( A d , A d \ { } ) be the universal maximal cover.Note that, by [8], a (( A d , A d \{ } ) = 1. Therefore there exist elements σ , . . . , σ in A d such that [ x σ , x σ ] and [ x σ , x σ ] in ([ F A d \{ } , F A d \{ } ] ∩ R A d ) / [ F A d \{ } , R A d ] have,respectively, order two and three.It is easy to see that for an element σ belonging to a conjugacy class C of type (I) thecentralizer Z ( σ ) ⊂ A d of the element σ is a cyclic group generated by σ . Therefore K C is the trivial group and hence s f ( C ) = h ( A d ). Therefore, by Proposition 4, a ( A d ,O ) = 6 if O contains only the elements of conjugacy classes of type (I).Let σ is of cycle type (2 , , σ = σ σ , where σ = (1 , ,
4) and σ = (5 , , Z ( σ ) ⊂ A d of σ is Kl ×h σ i , where Kl = h σ i×h σ i and σ = (1 , , x σ , x σ ,σ ± ] =[ x σ , x σ ] in the group F A d \{ } / [ F A d \{ } , R A d ]. Therefore K C , where C has type (2 , , σ is two (see Lemma 2) and it is oforder two if and only if [ x σ , x σ ] is not the unity in F A d \{ } / [ F A d \{ } , R A d ]. But, theembeddings h σ , σ i ⊂ A d ⊂ Σ d define a sequence of homomorphisms H ( h σ , σ i , Z ) → H ( A d , Z ) → H (Σ d , Z )such that the image of the non-trivial element [ x σ , x σ ] in H ( h σ , σ i , Z ) is a non-trivial in H (Σ d , Z ). Therefore s f ( C ) = 3 for the conjugacy class C of cyclic type(2 , ,
3) and, similarly, s f ( C ) = 3 for the conjugacy class C of cyclic type (2 , K C is a subgroup of H ( A d , Z ) ≃ Z / Z generated by the elements of the second order(see Proposition 3) and only the elements of K C and K C can generate the subgroupof order two in H ( A d , Z ).Let σ is of cycle type (3 , σ = σ σ , where σ = (1 , ,
3) and σ = (4 , , Z ( σ ) ⊂ A d of σ is h σ i × h σ i . Therefore [ x σ , x σ ] is not the unity in F A d \{ } / [ F A d \{ } , R A d ] only if σ = σ ± , either σ = σ ± , or σ = σ σ − , or σ = σ − σ . We have[ x σ σ − , x σ ] = [ x σ , x σ ][ x σ − , x σ ] = [ x σ , x σ ] in F A d \{ } / [ F A d \{ } , R A d ] and, similarly, [ x σ − σ , x σ ] = [ x σ , x σ ], since the elements x σ x − σ x − σ , x σ σ − x σ x − σ belong to the center of the group F A d \{ } / [ F A d \{ } , R A d ]. There-fore the group K C is a nontrivial group of order three if and only if K C is a nontrivialgroup of order three, where C is a conjugacy class of the cycle type (3) and C isthe conjugacy class of the cycle type (3 , s f ( C ) = s f ( C ) = 2. NowProposition 7 follows from Propositions 4 – 6. (cid:3) Cohomologycal description of the ambiguity indices
In notations used in Section 1, for an equipped finite group (
G, O ) a subgroup K ( G,O ) of H ( G, Z ) was defined as follows: K ( G,O ) is the subgroup of ( R O ∩ [ F O , F O ]) / [ F O , R O ]generated by the elements of R O of the form [ w, x g ], where g ∈ O and w ∈ F O , and k ( G,O ) is it’s order. HE AMBIGUITY INDEX OF AN EQUIPPED FINITE GROUP 17
Denote B ( G,O ) = K ⊥ ( G,O ) = { ϕ ∈ Hom ( H ( G, Z ) , Q / Z ) | ϕ ( w ) = 0 for all w ∈ K ( G,O ) } a subgroup of H ( G, Q / Z ) dual to K ( G,O ) . As in the proof of Theorem 1, it is easyto show that B ( G,O ) ) = ker[ H ( G, Q / Z ) → O A ⊂ G H ( A, Q / Z )] , where A runs over all abelian subgroups of G generated by two elements g ∈ O and h ∈ G . Let b ( G,O ) be the order of the group B ( G,O ) . In particular, b ( G,G \{ } ) = b ( G ).The next theorem immediately follows from Theorem 3. Theorem 10.
For an equipped finite group ( G, O ) we have a ( G,O ) = b ( G,O ) . The group H ( G, Q/Z ) is a direct sum of primary components H ( G, Q/Z ) =Σ p H ( G, Q/Z ) p where primes p run through a subset of primes dividing the order ofof H ( G, Q/Z ) and hence G . Therefore we have the following: Proposition 8.
If the set of conjugacy classes O consists of all classes of power ofprime order then a ( G,O ) = b ( G ) . Moreover it is sufficient to consider such classesonly for primes dividing h ( G ) . Note that H ( G, Q / Z ) p embeds into H ( Syl p ( G ) , Q / Z ) p where Syl p ( G ) is a Sy-low p -subgroup of G . Similarly the p -primary component B ( G ) p is a subgroup of B ( Syl p ( G )).More explicit versions of Proposition 8 for different groups provide with simplemethods to compute B O ( G )5. An example of a finite group G with b ( G ) > p -group G p which is a central extension of Z p = A p with generators x i . The center of G p is generated by pairwise commutators x i x j x − i x − j = [ x i , x j ] withone relation between [ x , x ][ x , x ] = 1. Thus there is natural exact sequence:1 → Z p → G p → A p → Lemma 6. ([1] , [13]) B o ( G p ) = Z /p .Proof. It is shown in [1] using standard spectral sequence that for a central extension G of an abelian group A the group B ( G ) is contained in the image of H ( A, Q / Z ) in H ( G, Q / Z ). The group H ( A p , Q / Z ) = Z p which is generated by elements [ x i , x j ] ∗ .The kernel of the map H ( A p , Q / Z ) → H ( G p , Q / Z ) is naturally dual to the center Z p of G p . Thus the image of H ( A p , Q / Z ) in H ( G p , Q / Z ) is a cyclic p -group generatedby one element w . Let us show that the latter is in B ( G p ). It is enough to checkthat it is trivial on any abelian subgroup in G p which surjects onto rank 2 subgroup Z p ⊂ Z p = A p . However G p does not contain such subgroups. Indeed assume thatthe restriction w on a subgroup with generators x , y ∈ G p is trivial. It means thatthe commutator [ x, y ] = 1 in G p where x, y are projections of x , y into A p . Onthe other hand the only nontrivial relation between commutators of elements in A p is[ x , x ][ x , x ] = 1 which is not equal to [ x, y ] for any pair x, y ∈ A p . Hence w restrictstrivially onto any subgroup with two generators in G p and generates B ( G p ). References [1] F. Bogomolov
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Courant Institute of Mathematical Sciences and National Research UniversityHigher School of Economics
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