The unitary cover of a finite group and the exponent of the Schur multiplier
aa r X i v : . [ m a t h . G R ] D ec The unitary cover of a finite group andthe exponent of the Schur multiplier.
Nicola Sambonet ∗ In memory of David Chillag.
Abstract
For a finite group we introduce a particular central extension, the unitarycover, having minimal exponent among those satisfying the projective liftingproperty. We obtain new bounds for the exponent of the Schur multiplierrelating to subnormal series, and we discover new families for which thebound is the exponent of the group. Finally, we show that unitary covers arecontrolled by the Zel’manov solution of the restricted Burnside problem for2-generator groups.
The
Schur multiplier of a finite group G is the second cohomology group withcomplex coe ffi cients, denoted by M( G ) = H ( G , C × ). It was introduced in thebeginning of the twentieth century by I. Schur, aimed at the study of projectiverepresentations. To determine M( G ) explicitly is often a di ffi cult task. Therefore,it is of interest to provide bounds for numerical qualities of M( G ) as the order, therank, and - our subject - the exponent.In 1904 Schur already showed that [exp M( G )] divides the order of the group,and this bound is tight as M( C n × C n ) = C n . Note that C n × C n is an example ofgroup satisfying exp M( G ) | exp G , (1)property which has been proven for many classes of groups. ∗ Department of Mathematics, Technion, Haifa, 32000 Israel, [email protected]. .1 Groups such that exp M(G) divides exp G Firstly, (1) holds for every abelian group G . Indeed, consider the cyclic decompo-sition ordered by recursive division: G = n ⊕ i = C d i , d i | d i + . (2)By Schur it is known (cf. [12, p. 317]) thatM( G ) = n ⊕ i = C d i n − i . (3)Consequently, exp M( G ) = d n − which in turn divides exp G = d n . A secondimportant example of groups enjoying (1) are the finite simple groups, whosemultipliers are known and listed in the Atlas [4].A standard argument (cf. [3, Th.10.3]) proposes to focus on p -groups. Indeed,the p -component of M( G ) is embedded in the multiplier of a p -Sylow via therestriction map. Therefore, if Π ( G ) denotes the set of prime divisors of G , and S p denotes a p -Sylow of G for p ∈ Π ( G ), thenexp M( G ) | Y p ∈ Π ( G ) exp M( S p ) . (4)Clearly, since exp G = Y p ∈ Π ( G ) exp S p , if (1) holds for every p -Sylow of G , then it does also for G .Then, a fundamental feature of p -groups is the nilpotency class. Recently,P. Moravec completed a result of M. R. Jones [11, Rem. 2.8] proving (1) forgroups of class at most 3, and extended this result to groups of class 4 in theodd-order case [17, Th. 12, Th. 13]. Moravec discovered many other familiesenjoying (1): metabelian groups of prime exponent [15, Pr. 2.12], 3-Engel groups,4-Engel groups in case the order is coprime with 2 and 5 [16, Cor. 5.5, Cor. 4.2], p -groups of class lower than p − p -groups of maximal class[18, Th. 1.4]. Without pretending to complete a list, we mention that (1) holdsfor extraspecial and abelian-by-cyclic groups, as follows from an argument ofR.J. Higgs [9, Pr. 2.3, Pr. 2.4].We have not cited 2-groups of class 4. Actually, the general validity of (1) wasdisproved by such a group long time before all the reported examples. A. J. Bayes,J. Kautsky and J. W. Wamsley introduced a group of order 2 and exponent 4whose multiplier has exponent 8 [1]. Lately, Moravec described another coun-terexample of order 2 and class 6 [15, Ex. 2.9]. Nevertheless, these essentially2re the only counterexamples we know: both were obtained by computer tech-nique and satisfy exp M( G ) = G = p > p -groups [13, Th. 2.4].By definition, a p -group G for p > powerful if the derived subgroup G ′ iscontained in the agemo subgroup ℧ ( G ) = h g p | g ∈ G i generated by the p -powers. Abelian p -groups are powerful, and if a p -group ispowerful, then its quotients of exponent p are necessarily abelian. For this reason,powerful p -group and groups of exponent p can be considered as two extremesdealing with p -groups for p > Beside the result on powerful p -groups, Lubotzky and Mann provided a bound forexp M( G ) involving the exponent and the rank of G [13, Pr. 2.6,Pr. 4.2.6], this hasbeen recently refined by J. Gonz´alez S´anchez and A. P. Nicolas [6, Th. 2].Meanwhile, Moravec proved the existence of a bound only in terms of theexponent of the group [15, Pr. 2.4]. This bound relies on the Zel’manov solution ofthe restricted Burnside problem, with the idea that the problem of finding boundsfor the exponent of the multiplier can be reduced in some extent to 2-generatorgroups, we will give an alternative proof of this fact.Since the use of the Zel’manov solution gives a bound which is apparently farfrom being e ffi cient, Moravec also stated a more practical bound:exp M( G ) | (exp G ) d − (5)where d is the derived length assumed to be greater than 1 [15, Th. 2.13]. Thebound analogue to (5) with the nilpotency class c in place of d was previouslydiscovered by Jones [11, Cor. 2.7], then modified asexp M( G ) | (exp G ) ⌈ c / ⌉ (6)by G. Ellis [5, Th. B1]. As Moravec illustrated, (5) improves (6) for c ≥
11 viathe formula d ≤ ⌊ log c ⌋ + Results
To give evidence for the content of this paper, we present some advancementconcerning the problems exposed in the introduction. We improve (5) and conse-quently (6), also including the case of abelian groups for which (1) holds.
Theorem A.
Let G be a p -group of derived length d . Thenexp M( G ) | ( d − · (exp G ) d p = G ) d p > Comparison with (5): in case p >
2, the bounds coincide for d = d >
2; in case p =
2, it is non-e ffi cient for d =
2, thebounds coincide for d = G =
4, and the improvement occurs in all theother cases.
Comparison with (6) via (7): in case p >
2, the bounds coincide for c = , , c =
8, and the improvement occurs for c = c ≥
9; in case p = G =
4, the bounds coincide for c =
7, and the improvement occurs for c ≥ p = G >
4, the bounds coincide for c =
7, and the improvementoccurs for c ≥ ff erence between the odd and the even case in Theorem A can be ex-plained with the concept of unitary cover ( § § § Theorem B.
There exists a canonical element Γ u ( G ), the unitary cover of G ,which has minimal exponent in the set of central extensions of G satisfying theprojective lifting property. The map Γ u , associating to a group its unitary cover,satisfies for any normal subgroup N of G the following properties:i) exp M( G ) | exp Γ u ( N ) · exp M( G / N )ii) exp Γ u ( G ) | exp Γ u ( N ) · exp Γ u ( G / N )iii) Γ u ( G / N ) is a homomorphic image of Γ u ( G ) .Moreover, if G = N ⋊ H , theniv) exp M( G ) | lcm { exp Γ u ( N ) , exp M( H ) } v) exp Γ u ( G ) | lcm { exp Γ u ( N ) · exp H , exp Γ u ( H ) } .By minimality, one can eventually replace the unitary cover with any centralextension satisfying the projective lifting property, for instance with any Schurcover. The word “canonical” refers to the fact that Γ u ( G ) is uniquely defined,whereas two Schur covers need not to be isomorphic.4e determine the exponent of the unitary cover for abelian p -groups, andin case p > p -groups (introduced in § Lemma C.
The following holds.i) If G is a powerful p -group for p >
2, then exp Γ u ( G ) = exp G .ii) If G is an abelian 2-group of exponent n . Thenexp Γ u ( G ) = σ · exp G for σ = ( G has a subgroup isomorphic with C n × C n Corollary D.
Let G be a p -group, and N a normal subgroup of G . Assume one ofthe following: p = N is abelian with no subgroups isomorphic with C k × C k ,where 2 k is the exponent of G . p > N is a powerful p -group . And assume one of the following: ( M( G / N ) = . G = N ⋊ H where H satisfies (1).Then G satisfies property (1).At least for groups of odd order, the previous result generalizes the case ofabelian-by-cyclic groups to powerful-by-trivial multiplier groups, and it reveals aclosure property under semidirect products with powerful kernels.Our next result concerns regular p -groups, which constitute one of the mostimportant family of p -groups and were introduced by P. Hall in 1934 [7, § p -group G is regular if for every x , y ∈ G there exist c ∈ h x , y i ′ such that( xy ) p = x p y p c p . Abelian p -groups are regular, regular 2-groups are abelian, andregular p -groups share important properties with abelian groups for any p .Many families of groups for which (1) has been proven consist of regular p -groups, at first abelian p -groups and p -groups of class lower than p (cf. [2, p. 98]and § P = G / ℧ ( G ) belongs to some of such classes, then G is regular and it also satisfies (1). We refer to the first Hall criterion claimingthat if | P / ℧ ( P ) | < p p , then G is regular and it is said absolutely regular . Proposition E. If G is a regular p -group and exp M( G / ℧ ( G )) divides p , then G satisfies (1). Moreover, (1) holds for groups of exponent p i ff it holds for regular p -groups. In particular, absolutely regular p -groups enjoy this property, and ingeneral regular 3-groups. 5e shall now prove the bound concerning the derived length, since we ob-tain the result in its stronger versions, first involving any subnormal series, theninvolving abelian 2-groups and powerful p -groups for p > Proof of Theorem A.
By iteration of Theorem B, if a group G admits a subnor-mal series G = G > G > · · · > G r − > G r = , G i E G i − , Q i = G i − / G i , (8)then exp Γ u ( G ) | r Y j = exp Γ u ( Q i ) . By Lemma C, we have respectively:I. Let G be a p -group for p >
2. Assume G admits a subnormal series (8) where Q i are powerful p -groups. Thenexp M( G ) | r Y j = exp Q j · exp M( Q ) . II. Let G be a 2-group. Assume G admits a subnormal series (8) where Q i areabelian. Then exp M( G ) | | I | · r Y j = exp Q j · exp M( Q ) . where I ⊆ { , . . . , r } is such that k ∈ I i ff Q k has a subgroup isomorphic with C e k × C e k for e k = exp Q k .These bounds prove Theorem A considering the derived series, so that thefactor Q i are abelian. In case p > p -groups are powerful. Incase p = | I | with d −
1. Notice that exp Q k divides exp G for every k , as well as exp M( Q ) divides exp G since Q = G / G ′ is abelian. (cid:3) We expose our alternative proof that the study of the exponent of the multipliercan be restricted in some extent to 2-generator groups.
Proposition F.
Let S ( G ) denote the set of 2-generator subgroups of G , thenexp Γ u ( G ) | lcm S ∈ S ( G ) exp Γ u ( S ) . For a fixed positive integer n , let S ( n ) denote the set of isomorphism classes of2-generator groups whose exponent divides n . Substituting S ( G ) with S (exp G )in the bound of Proposition F, we obtain a bound depending only on the exponentof the group (cf. [15, Pr. 2.4]). 6e assume that exp G = p k , by the Zel’manov solution of the restricted Burn-side problem [21], [22], there exists a finite group B p k = RBP(2 , p k )such that every element in S ( p k ) is a homomorphic image of B p k . Therefore, byTheorem B we have that exp Γ u ( S ) divides exp Γ u ( B p k ) for every S in S ( p k ), andwe can also add some information to this result. Proposition G. If G is a group of exponent p k , thenexp Γ u ( G ) | exp Γ u ( B p k ) . Moreover, exp Γ u ( B p k ) = p k · exp M( B p k )and p k | exp M( B p k ) . Given an account on the theory of central extensions ( § § § § § Let G be a group, and A an abelian group. A is a map α : G × G → A satisfying α ( x , y ) · α ( xy , z ) = α ( x , yz ) · α ( y , z ) . A is a cocycle obtained from a map ζ : G → A as δζ ( x , y ) = ζ ( x ) · ζ ( y ) · ζ ( xy ) − . The sets of cocycles and coboundaries are denoted with Z ( G , A ) and B ( G , A )respectively, they constitute abelian groups under pointwise multiplication. Thequotient H ( G , A ) = Z ( G , A ) / B ( G , A ) is the second cohomology group . In theparticular case A = C × , we obtain the Schur multiplier M( G ) = H ( G , C × ), andwe briefly denote Z ( G ) = Z ( G , C × ) and B ( G ) = B ( G , C × ).These definitions play a fundamental role in the theory of central extensions,and in the theory of projective representations. We will give an account hereby,recommending the reading of [10, pp.181-185]. Accordingly with this referencewe adopt the right notation x y = y − xy and [ x , y ] = x − y − xy .7 .1 Central extensions and Schur covers A central extension of a group G is a group Γ having a central subgroup A ≤ Z ( Γ )such that Γ / A is isomorphic with G . It is usually written as ω : 1 → A → Γ π → G → A = ker π , and ω will be now defined. Let φ : G → Γ be a section, that is π ( φ ( g )) = g for every g ∈ G . By definition, every γ ∈ Γ can be uniquely written as γ = a · φ ( g ) for some a ∈ A and some g ∈ G . Then, ω : G × G → A is associatedwith φ by the relation φ ( g ) · φ ( h ) = ω ( g , h ) · φ ( gh ) , and in turn ω ∈ Z ( G , A ).Consider now a di ff erent section φ ′ : G → Γ , clearly φ ′ ( g ) = ζ ( g ) · φ ( g )for some ζ : G → A . From the analogue relation defining ω ′ , it follows that ω ′ = ω · δζ . Multiplication by a coboundary correspond to a change of section.We may also mention that the trivial cocycle G × G → { } ≤ A corresponds to thetrivial extension Γ = G × A .We briefly show how the Schur multiplier parametrizes the central extensions.Denote by ˇ A = Hom( A , C × ) the group of the irreducible characters of A . Thenthere exists η : ˇ A → M( G ) called the standard map , defined as η : ˇ A → M( G ) , λ η ( λ ) = [ λ ◦ ω ] , λ ◦ ω ( x , y ) = λ ( ω ( x , y )) . By the discussion above η is well-defined, and it is easy to see that η is a homo-morphism such thatker η = ( A ∩ Γ ′ ) ⊥ , ( A ∩ Γ ′ ) ⊥ = { χ ∈ ˇ A | A ∩ Γ ′ ≤ ker π } . The standard map also leads to the definition of Schur covers: a central exten-sion is a
Schur cover of G if the standard map is an isomorphism. An equivalentdefinition is the following: a Schur cover of G is a central extension such that thekernel is isomorphic with the Schur multiplier and it is contained in the derivedsubgroup, 1 → M → Γ → G → , M ≃ M( G ) , M ≤ Z ( Γ G ) ∩ Γ ′ G . If we make the weaker assumption that the standard map is onto, then Γ has the projective lifting property . This is equivalent to the following property,1 → A → Γ → G → , A ∩ Γ ′ ≃ M( G ) , A ≤ Z ( Γ G ) . If Γ has the projective lifting property, then exp M( G ) has to divide exp Γ .Therefore, it has interest to find a minimal bound for the exponent of an extensionswith the projective lifting property. We remark that this lower bound has not to berealized by a Schur cover, as shown by the following example.8 xample 1. Consider the semidirect product of two cyclic groups of order p defined by G = h x , y | x p = y p = , y x = y p + i . It can be seen, for instance using
Gap [23], that exp M( G ) = p and that the group Γ = h ¯ x , ¯ y | ¯ x p = ¯ y p = , ¯ y ¯ x = ¯ y p + i is the only Schur cover of G , and it has exponent p . Nevertheless, the group Γ = h ˜ x , ˜ y , ˜ z | ˜ x p = ˜ y p = ˜ z p = , ˜ y ˜ x = ˜ y p + · ˜ z , [˜ z , ˜ x ] = [˜ z , ˜ y ] = i has the projective lifting property for G , and satisfies exp Γ = p .Among the central extensions those with the projective lifting property havefundamental importance, as they permit to transfer results on ordinary represen-tations to projective representations and vice-versa. This was depicted by Schurwho also proved by a constructive method that Schur covers always exist ( § µ ∈ M( G ) we will say that a central extension1 → A µ → Γ → G → µ -cover if A µ is cyclic and the standard map η µ maps ˇ A µ onto h µ i . Thisdefinition is not usually stated, as for any µ ∈ M( G ) it is possible to obtain a µ -cover as a quotient of any extension with the projective lifting property. Proposition 2.
Let Γ be an extension of G with the projective lifting property, and µ ∈ M( G ). Then a µ -cover Γ µ can be obtained as a quotient of Γ G . In particular,exp Γ µ divides exp Γ G . Proof.
Since the standard map η G : ˇ A → M( G ) is assumed to be onto, there existsa preimage λ ∈ ˇ A of µ under η G , that is η G ( λ ) = µ . We claim that the µ -cover is Γ µ = Γ G / ker λ , whose exponent divides exp Γ G . Set A µ = A / ker λ , then λ can beidentified with a faithful irreducible character λ µ of the cyclic group A µ , and thestandard map η µ : ˇ A µ → h µ i is onto. (cid:3) We write down some complementary formulas for further reference. Let Γ beany central extension. For any section φ , an element γ of Γ is uniquely written as γ = a · φ ( g ) for some a ∈ A and some g ∈ G . Hence, o ( γ ) divides lcm { o ( a ) , o ( φ ( g )) } and since o ( φ ( g )) = o ( g ) · o ( φ ( g ) o ( g ) ) it holdsexp Γ = lcm { exp A , max g ∈ G o ( g ) · o ( φ ( g ) o ( g ) ) } . (9)9oreover, for g ∈ G it is not di ffi cult to see that φ ( g ) o ( g ) = o ( g ) − Y j = ω ( g , g j ) . (10)Finally, concerning conjugation in Γ , by comparison of φ ( x ) · φ ( y ) and φ ( y ) · φ ( x y )it follows that φ ( x ) φ ( y ) = ω ( x , y ) ω ( y , x y ) − · φ ( x y ) (11)holds for any x , y ∈ G . Proposition 3.
Let Γ be a central extension of a d -generator group G . Then thereexists a d -generator sugroup X of Γ , which is a central extension of G such that X ′ = Γ ′ . In particular, if Γ has the projective lifting property, then also X does,and if Γ is a Schur cover, then X = Γ . Proof.
Let G = h x , . . . , x d i and φ : G → Γ be any section. We claim that thedesired subgroup is X = h φ ( x ) , . . . , φ ( x d ) i . For g ∈ G fix a writing g = x ε i · · · x ε l i l ,then φ ( g ) = b · φ ( x i ) ε · · · φ ( x i l ) ε l = b · ξ , for b ∈ A and ξ ∈ X . Any γ ∈ Γ isuniquely written as a · φ ( g ) for some a ∈ A and g ∈ G . Therefore, since A ≤ Z ( Γ ),then Γ ′ = h [ γ , γ ] | γ i ∈ Γ i = h [ ξ , ξ ] | ξ i ∈ X i = X ′ . (cid:3) In analogy to the group algebra C [ G ] for ordinary representations, for projectiverepresentations it is defined the twisted group algebra , which in turn relies on thecocycles. For α ∈ Z ( G ), C α [ G ] is the C -algebra with basis ¯ G = { ¯ g | g ∈ G } identified with the group, and product ¯ x · ¯ y = α ( x , y ) · xy obeying to the groupproduct unless a twisting coe ffi cient.The cocycle condition is the associative law ( ¯ x · ¯ y ) · ¯ z = ¯ x · (¯ y · ¯ z ), whereasmultiplication by a coboundary represents a locally-linear change of group-basis˜ g = ζ ( g ) · ¯ g . As common we consider normalized cocycles, that is α (1 , =
1. Themeaning of this assumption is that ¯1 is the identity of the twisted group algebra.Hence, for normalized coboundaries δζ it can be assumed ζ (1) = H of G , the restriction map is defined byres : M( G ) → M( H ) , [ α ] [ α H ] , α H ( h , h ) = α ( h , h ) , and there is a natural identification C α H [ H ] ≤ C α [ G ]. Then, for a normal subgroup N of G the inflation map is defined byinf : M( G / N ) → M( G ) , [ β ] [ β ∗ ] , β ∗ ( g , g ) = β ( g N , g N ) . M ( G / N ) is contained in the kernelof the restriction to M( N ), a description of these subgroups can be done in termsof the idempotents of C α N [ N ]. We recall that the twisted group algebra C α [ G ] issemi-simple: it admits a decomposition in irreducible subspaces, each one definedby an idempotent.It can be seen that [ α H ] = ff C α H [ H ] admits a 1-dimensional idempotent.Moreover, for a normal subgroup N of G , it was proven by R. J. Higgs [8, Pr. 1.5]that [ α ] = [ β ∗ ] for some β ∈ Z ( G / N ) i ff C α N [ N ] admits a 1-dimensional idem-potent which is invariant under conjugation in C α [ G ]. In analogy to (11), for any x , y ∈ G comparing ¯ x · ¯ y and ¯ y · x y we have the relation¯ x ¯ y = α ( x , y ) · α ( y , x y ) − · x y (12)which describes conjugation in C α [ G ]. Proposition 4.
Let N E G and α ∈ Z ( G ). If α N = α ( n , g ) = α ( g , n g ) forevery n ∈ N and g ∈ G , then [ α ] is inflated from G / N . If in addition G = N ⋊ H and [ α H ] =
1, then it holds [ α ] = Proof.
Since α N =
1, then C α N [ N ] admits the principal idempotent ε N = | N | X n ∈ N ¯ n , which is invariant in C α [ G ] by (12). In case G = N ⋊ H , since we assume [ α H ] = C α H [ H ] admits a central 1-dimensional idempotent υ H . As in the generalcase, C α N [ N ] admits the principal idempotent ε N . By (12) ε N and υ H commutes,so that ε N · υ H is a 1-dimensional idempotent of C α [ G ], and [ α ] = (cid:3) Also for the powers there is a formula analogue to (10), as for any g ∈ G itholds ¯ g o ( g ) = o ( g ) − Y j = α ( g , g j ) . (13)Cocycles whose group-basis satisfy the identity ¯ g o ( g ) = g ∈ G will playthe main role in the next section. We abstract the fundamental tool for our main results. We give a generalizationof the construction which proves Schur’s theorem on the existence of a coveringgroup, this will lead to the definition of the unitary covers ( § efinition 5. Let H be a finite subgroup of Z ( G ). We define a central extension1 → ˇ H → ˇ H ∝ G → G → , ˇ H ≤ Z ( ˇ H ∝ G ) . The underlying set of ˇ H ∝ G is G × H , and multiplication is given by the rule( g , χ ) · ( h , ψ ) = ( gh , ω ( g , h ) · χψ ) , where ω ( g , h ) ∈ ˇ H is defined by ω ( g , h )( α ) = α ( g , h ) for α ∈ H .The proof of Schur’s theorem is done in this terms: since B ( G ) is a divisiblesubgroup of finite index in Z ( G ), then it has a complement Z ( G ) = B ( G ) ⊕ J ,and ˇ J ∝ G is a Schur cover of G (cf. [10, Th. 11.17]).This construction is natural respect to the standard map in the following sense.For a cyclic decomposition H = h α i ⊕ . . . ⊕ h α k i , the dual group admits thedecompositionˇ H = h ˇ α i ⊕ . . . ⊕ h ˇ α k i , ˇ α i ( α j ) = ( e πι/ o ( α i ) if j = i , ι = √− , and there is a canonical identification of H with the double dual H ˇˇ ≡ H , α i ˇˇ ≡ α i , under this identification the standard map relative to ˇ H ∝ G is the projection from Z ( G ) to M( G ) η : H ˇˇ ≡ H → M( G ) , η ( α ) = [ α ] . Referring to § Lemma 6.
The extension ˇ H ∝ G has the projective lifting property i ff everycocycle in Z ( G ) is cohomologous with a cocycle of H , and it is a Schur cover i ff in addition H ∩ B ( G ) = K ≤ H ≤ Z ( G , A ), there is a natural isomorphismˇ K → ˇ H / K ⊥ , K ⊥ = { χ ∈ ˇ H | K ≤ ker χ } , defined choosing coherent cyclic decompositions ( K = h β i ⊕ . . . ⊕ h β l i H = h α i ⊕ . . . ⊕ h α l i ⊕ . . . ⊕ h α k i , β i ∈ h α i i , then setting ˇ β i ˇ α i K ⊥ , this induces an isomorphismˇ K ∝ G ≃ ( ˇ H ∝ G ) / K ⊥ . A case of particular interest is when K is obtained via the inflation map.12 emma 7. Let N be a normal subgroup of G , H be a finite subgroup of Z ( G ),and L be a finite subgroup of Z ( G / N ) such that H ∩ Z ( G / N ) ∗ = H ∩ L ∗ . Denote K = H ∩ L ∗ , ˙ N = h ( n , H ) | n ∈ N i , then there is a surjection ˇ L ∝ ( G / N ) ։ ( ˇ H ∝ G ) / K ⊥ ˙ N , which is an isomorphism in case L ∗ ≤ H . Proof.
If (1 , χ ) ∈ ˙ N ∩ ˇ H , then χ = Q kj = ω ( n , j , n , j ) for some n i , j ∈ N , so that χ ∈ K ⊥ . Consequently, ˙ N ∩ ˇ H ≤ K ⊥ . Also, ( n , H ) ( g ,χ ) = ( n g , ω ( n , g ) ω ( g , n g ) − ) and ω ( n , g ) ω ( g , n g ) − ∈ K ⊥ . Therefore, K ⊥ ˙ N E ˇ H ∝ G . Therefore, we obtain the central extension1 → ˇ H / K ⊥ → ( ˇ H ∝ G ) / K ⊥ ˙ N → G / N → H ∝ G ) / K ⊥ ˙ N is a homomorphic image of ˇ L ∝ ( G / N ). Write L = h γ i ⊕ . . . ⊕ h γ l i ⊕ . . . ⊕ h γ m i K = h β i ⊕ . . . ⊕ h β l i , β i ∈ h γ ∗ i i H = h α i ⊕ . . . ⊕ h α l i ⊕ . . . ⊕ h α k i , β i ∈ h α i i . If β i = ( γ ∗ i ) m i , then there is an isomorphismˇ L / ˜ K ⊥ ≃ ˇ K , ˇ γ i ˜ K ⊥ ˇ β i , ˜ K = h γ m i ⊕ . . . ⊕ h γ m l l i , which can be composed with the canonical isomorphism ˇ K ≃ ˇ H / K ⊥ to giveˇ L ։ ˇ L / ˜ K ⊥ ≃ ˇ H / K ⊥ , ˇ γ i ˇ α i K ⊥ . Then, observe that ( gn , H ) = ( g , H ) · ( n , ω ( g , n ) − ) and that ( n , ω ( g , n ) − ) ∈ K ⊥ ˙ N .The map ( gN , ˇ γ k · · · ˇ γ k m m ) ( g , ˇ α k · · · ˇ α k l l ) K ⊥ ˙ N is well defined, and it is the desired homomorphism. In case L ∗ ≤ H , then K = L ∗ and ˇ L ≃ ˇ K . Thus, the map described is one to one. (cid:3) .2 Unitary cocycles and unitary covers We introduce a subgroup of Z ( G ), whose definition is done accordingly to (13),and we mimic the Schur’s construction ( § Definition 8.
A cocycle α ∈ Z ( G ) is said to be unitary if o ( g ) − Y j = α ( g , g j ) = g ∈ G . The set of unitary cocycles constitutes a group denoted by Z u ( G ),the unitary cover of G is the extension Γ u ( G ) = Z u ( G )ˇ ∝ G . Unitary cocycles and unitary covers are the core of our main results. We beginproving that every cocycle is cohomologous with an unitary, so that cohomologycan be done with unitary cocycles exclusively. We describe the unitary cobound-aries and provide a relation wich refers to conjugation as shown by (12).At once we will show a clear benefit of these facts, as we ready give the firststatement of Theorem B in its explicit formulation.
Lemma 9.
Let α ∈ Z ( G ). Then:i) There exists β ∈ Z u ( G ) such that [ α ] = [ β ]. Therefore, M ( G ) ≃ Z u ( G ) / B u ( G )where B u ( G ) = B ( G ) ∩ Z u ( G )is the group of unitary coboundaries.ii) If δζ ∈ B u ( G ) for ζ : G → C × , then ζ ( g ) o ( g ) = g ∈ G .iii) Z u ( G ) and B u ( G ) are finite, and exp B u ( G ) divides exp G .iv) If β ∈ Z u ( G ), then β ( x , g ) o ( x ) = β ( g , x g ) o ( x ) for every x , g ∈ G . Proof. i) let α be any cocycle, define ξ ( g ) to be any o ( g )-root of Q o ( g ) − j = α ( g , g j ),set β = α · δξ − , then β is the unitary cocycle cohomologous with α . ii) applythe definition of unitary cocycle to δζ . iii) follows from ii. iv) in C β [ G ] it holds( ¯ x ¯ g ) o ( x ) = ( β ( x , g ) β ( g , x g ) − · x g ) o ( x ) = [ β ( x , g ) β ( g , x g ) − ] o ( x ) · ( x g ) o ( x ) by (12). Since β ∈ Z u ( G ), then ( ¯ x ¯ g ) o ( x ) = ( ¯ x o ( x ) ) ¯ g =
1, and since o ( x g ) = o ( x ), then ( x g ) o ( x ) = β ( n , g ) β ( g , n g ) − ] o ( x ) = (cid:3) emma 10. The exension Γ u ( G ) has the projective lifting property for G , and itsatisfies exp Γ u ( G ) = lcm { exp Z u ( G ) , exp G } . Moreover, if Γ is a central extension of G having the projective lifting property,then exp Γ u ( G ) divides exp Γ . Proof.
That Γ u ( G ) has the projective lifting property it follows by Lemma 9 andLemma 6. To find the exponent we use (9) with the section φ ( g ) = ( g , A ), where A = Z u ( G ), by definition φ ( g ) o ( g ) = ( g o ( g ) , o ( g ) − Y j = ω ( g , g j )) = Local-version.
Let µ ∈ M( G ), and Γ µ be a µ -cover. If β ∈ Z u ( G ) is such that µ = [ β ], then o ( β ) divides exp Γ µ . Step I . It is enough to prove that there exists one cocycle β with the requiredassertion, since two such cocycles di ff er by an unitary coboundary, and by Lemma9 unitary coboundaries have order dividing exp G thus dividing exp Γ µ . Since Γ µ is a µ -cover, the standard map η µ : ˇ A µ → h µ i is onto. Therefore, there exists λ ∈ ˇ A µ such that η µ ( λ ) = µ , where η µ ( λ ) = [ λ ◦ ω µ ]. Reading the proof ofLemma 9 together with (10), an unitary cocycle β cohomologous to λ ◦ ω µ isfound defining ζ : G → C × to be for g ∈ G any o ( g )-root of λ ( φ ( g ) o ( g ) ), thensetting β = ( λ ◦ ω µ ) · δζ − . We show the assertion by use of (9). Since o ( ω µ )divides exp A µ and λ is a homomorphism, then the order of λ ◦ ω µ divides exp A µ ,and since λ is faithful, then o ( λ ( φ ( g ) o ( g ) )) = o ( φ ( g ) o ( g ) ). Therefore, o ( ζ ( g )) = o ( g ) · o ( λ ( φ ( g ) o ( g ) )) = o ( g ) · o ( φ ( g ) o ( g ) ) , and o ( δζ ) divides max g ∈ G o ( g ) · o ( φ ( g ) o ( g ) ). Thus, o ( β ) divides lcm { o ( λ ◦ ω µ ) , o ( δζ ) } which divides lcm { exp A µ , max g ∈ G o ( g ) · o ( φ ( g ) o ( g ) ) } that by (9) is exp Γ µ . Step II.
For µ ∈ M( G ), by Lemma 9 there exists β µ ∈ Z u ( G ) such that [ β µ ] = µ .By Proposition 2 there exists a µ -cover Γ µ obtained as a quotient from Γ G , thenby the local-version o ( β µ ) divides exp Γ µ and consequently exp Γ G . In particular,defining J = h β µ | µ ∈ M( G ) i , then J is a subgroup of Z u ( G ) of exponent dividingexp Γ G . Clearly Z u ( G ) = JB u ( G ), so that exp Z u ( G ) = lcm { exp J , exp B u ( G ) } which divides lcm { exp Γ G , exp B u ( G ) } . The proof is complete by Lemma 9 sinceexp B u ( G ) divides exp Γ G . (cid:3) .3 Proof of the main theorems Proof of Theorem B.
The first statement is part of Lemma 10. The proof of i)and iv) is explicitly written while proving ii) and v), nevertheless, we shall givean independent proof only based on commonly known results. i) We prove thatexp M( G ) | lcm { exp Z u ( N ) , exp N } · exp M( G / N ) , then we apply Lemma 10. For any co-class [ α ] ∈ M( G ), we can assume by Lemma9 that α ∈ Z u ( G ). For r = lcm { exp Z u ( N ) , exp N } , we show that α r satisfies the firsttwo conditions of Proposition 4, proving that [ α ] r is inflated from M( G / N ) so thatit becomes trivial when risen to the exp M( G / N )-power. Since exp Z u ( N ) divides r clearly ( α N ) r =
1, and since exp N divides r by Lemma 9 the proof is complete. iv) We assume G = N ⋊ H , and we prove thatexp M( G ) | lcm { exp Z u ( N ) , exp N , exp M( H ) } , then we apply Lemma 10. By a result of K. Tahara [20, Th. 2], M( G ) is isomorphicwith the direct sum of M( H ) and the kernel of the restriction from M( G ) to M ( H ).Denote by r be the lcm, since exp M( H ) divides r we can consider only co-classes[ α ] ∈ M( G ) whose restriction to H is trivial. By Lemma 9 we can also assumethat α ∈ Z u ( G ). The third condition of Proposition 4 is immediate, and sincelcm { exp Z u ( N ) , exp N } divides r the first two conditions follow the general case. ii) We prove thatexp Z u ( G ) | lcm { exp Z u ( N ) , exp N } · lcm { exp Z u ( G / N ) , exp G / N } , then we apply Lemma 10. Fix α ∈ Z u ( G ) and a transversal T for N in G . Define ξ ( g ) = α ( t , n ) for g = tn where t ∈ T and n ∈ N . Let r = lcm { exp Z u ( N ) , exp N } ,define β = α r and ˜ β = β · δϑ for ϑ = ξ r . We show that ϑ ( g ) o ( gN ) = g ∈ G , and that ˜ β is inflated from Z u ( G / N ), as o ( β ) divides lcm { o ( ˜ β ) , o ( δϑ ) } thiscompletes the proof. For g , g ∈ G , let g i = t i n i where t i ∈ T and n i ∈ N , andlet t t = t , n , where t , ∈ T and n , ∈ N . In the twisted group algebra C β [ G ]consider ϑ ( g ) o ( gN ) · ¯ g o ( gN ) = ( ϑ ( g ) · ¯ g ) o ( gN ) = (¯ t · ¯ n ) o ( gN ) = ¯ t o ( gN ) · ¯ n ¯ t o ( gN ) − · · · ¯ n ¯ t · ¯ n . For any x ∈ G since ¯ x o ( x ) = ( x o ( xN ) ) o ( x o ( xN ) ) =
1, then o ( xN ) − Y j = α ( x , x j ) o ( x o ( xN ) ) = , x ∈ G ,
16n particular ¯ g o ( gN ) = g o ( gN ) and ¯ t o ( gN ) = t o ( gN ) as tN = gN . Since exp N divides r ,by Lemma 9 then ¯ n ¯ t j = n t j for every j , and since exp Z u ( N ) divides r , then β N = t o ( gN ) · ¯ n ¯ t o ( gN ) − · · · ¯ n ¯ t · ¯ n = t o ( gN ) · n t o ( gN ) − · · · n t · n = g o ( gN ) proving that ϑ ( g ) o ( gN ) =
1. We now prove that ˜ β = γ ∗ for some γ ∈ Z u ( G / N ).Notice that ξ ( t ) = ξ ( n ) = t ∈ T and any n ∈ N , so that α ( g , g ) · δξ ( g , g ) = α ( t , t ) · δξ ( t , t ) ·· [ α ( n , t ) · α ( t , n t ) − ] · α ( n , , n t n ) · α ( n t , n ) . Hence, define γ ∈ Z ( G / N ) by setting γ ( g N , g N ) = ˜ β ( g , g ), since o ( gN ) − Y j = ˜ β ( g , g j ) = o ( gN ) − Y j = α ( g , g j ) r · δϑ ( g , g j ) = o ( gN ) − Y j = δϑ ( g , g j ) = ϑ ( g ) o ( gN ) = , then it holds γ ∈ Z u ( G / N ). v) Assume G = N ⋊ H , and choose the transversal T = H so that α ( g , g ) · δξ ( g , g ) = α ( h , h ) · [ α ( n , h ) · α ( h , n h ) − ] · α ( n h , n ) . The proof that o ( ξ ( g )) divides exp Γ u ( N ) · o ( gN ) follows the general case, and thisgives the bound exp Z u ( G ) | lcm { exp Z u ( H ) , exp Γ u ( N ) · exp H } where Γ u can replace Z u by Lemma 10 with no loss. iii) Since Z u ( G / N ) ∗ ≤ Z u ( G ), then Γ u ( G / N ) ≃ Γ u ( G ) / ( Z u ( G / N ) ∗ ) ⊥ ˙ N by Lemma 7. (cid:3) Proof of Lemma C.
We begin finding a cover of minimal exponent for an abelian p -group A . Write the cyclic decomposition (2), then Γ = h x , . . . , x m | o ( x i ) = p d i , [[ x i , x j ] , x k ] = i is a cover for A (cf. [12, p. 325]), which satisfies exp Γ = exp A for p >
2, andexp
Γ = σ · exp A for p =
2. It can be seen that exp Z u ( A ) = exp Γ , and by Lemma10 it follows that exp Γ u ( A ) = exp Γ . The case p = p > p -groups. By definition, G ′ ≤ ℧ ( G ) sothat G / ℧ ( G ) is elementary abelian, and we can assume that exp G > p . Sinceexp Γ u ( G ) = exp Γ u ( ℧ ( G )) · exp Γ u ( G / ℧ ( G )), by Theorem B, the result follows byinduction as ℧ ( G ) is powerful and exp ℧ ( G ) = exp G / p [13, Cor. 1.5, Pr. 1.7]. (cid:3) roof of Proposition E. We can assume that exp G ≥ p >
2. It is known that ℧ ( G ) is powerful [13, p. 497], by Theorem B and Lemma C, then exp M( G ) di-vides exp ℧ ( G ) · exp M( G / ℧ ( G )). Moreover, every element of ℧ ( G ) is a p -power[2, Th. 7.2], so that exp ℧ ( G ) = exp G / p . We remind that exp G / ℧ ( G ) = p , andgroups of exponent p are regular. Assuming that exp M( G / ℧ ( G )) divides p theproof is completed. This is the case for p =
3, as well for absolutely regular p -groups as cl( G / ℧ ( G )) < p . (cid:3) Proof of Proposition F.
Since there exists a R ∈ S ( G ) such that exp R = exp G ,by Lemma 10 it is enough to prove that there exists S ∈ S ( G ) such that exp Z u ( G )divides exp Z u ( S ). Choose any element µ ∈ M( G ) satisfying o ( µ ) = exp M( G ).By Lemma 9, there exists α ∈ Z u ( G ) such that µ = [ α ]. Let x , y ∈ G such that o ( α ) = o ( α ( x , y )), and set S = h x , y i . Since α S ∈ Z u ( S ) and o ( α S ) = o ( α ), then o ( µ ) divides o ( α ) and the proof is complete. (cid:3) Proof of Lemma G.
For any p -group G and any integer m , the m -agemo sub-group is defined as ℧ m ( G ) = h g p m | g ∈ G i . Let Γ be any central extension of B p k , by Proposition 3 there exists a 2-generatedsubgroup X of Γ which is a central extension of B p k such that Γ ′ = X ′ . Since B p k is the maximal 2-generated group of exponent p k , it follows that X / ℧ k ( X ) ≃ B p k .Therefore, A ∩ Γ ′ = A ∩ X ′ ≤ ℧ k ( X ). The assertionexp Γ u ( B p k ) = p k · exp M( B p k )follows, since Γ u ( B p k ) has the projective lifting property.We now prove that p k divides exp M( B p k ). For any integer l , the sequence1 → ℧ k ( B p k + l ) → B p k + l → B p k → → ℧ k ( B p k + l ) / [ ℧ k ( B p k + l ) , B p k + l ] → B p k + l / [ ℧ k ( B p k + l ) , B p k + l ] → B p k → . There is an embedding ( § ℧ k ( B p k + l ) ∩ B ′ p k + l / [ ℧ k ( B p k + l ) , B p k + l ] ֒ → M( B p k ) , which for large enough l becomes an isomorphism, and the exponent of this groupis p m for the minimal m such that ℧ m ( ℧ k ( B p k + l ) ∩ B ′ p k + l ) ≤ [ ℧ k ( B p k + l ) , B p k + l ] . G of exponent p k + l is a homomorphic image of B p k + l , then ℧ m ( ℧ k ( G ) ∩ G ′ ) ≤ [ ℧ k ( G ) , G ] , therefore, lower bounds for exp M( B p k ) derives from two generators groups. Con-sider the covering group of C n × C n defined by G = h x , y | x p k = y p k = , y x = y p k + i , then G ′ = ℧ k ( G ) = Z ( G ) = h y k i , so that ℧ m ( ℧ k ( G ) ∩ G ′ ) = ℧ m ( h y k i ) , [ ℧ k ( G ) , G ] = , so that m ≥ k completing the proof. (cid:3) Acknowledgement
The author is conducing his PhD studies, he is indebted to his menthors Prof. EliAljade ff (Technion) and Dr. Yuval Ginosar (University of Haifa). A preliminaryversion was presented at ”Groups St Andrews 2013”, University of St Andrews,Scotland. References [1] A. J. Bayes, J. Kautsky, J. W. Wamsley.
Computation in nilpotent groups(application) . Proc. 2nd Internat. Conf. Theory of Groups, Lecture Notesin Math. 372 (1973), Springer Verlag, 82–89.[2] Y. Berkovich.
Groups of Prime Power Order, Vol. I . De Gruyter, 2008.[3] K. S. Brown.
Cohomology of Groups . Springer Verlag, Berlin, 1982.[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson.
ATLASof Finite Groups . Clarendon Press, Oxford, 1985.[5] G. Ellis.
On the relation between upper central quotients and lower centralseries of a group . Trans. Amer. Math. Soc. 353 (2001), no. 10, 4219–4234.[6] J. Gonz´alez-S´anchez and A. P. Nicolas.
A bound for the exponent of theSchur multiplier of a finite p-group . J. Algebra 324 (2010), 2564–2567.[7] P. Hall.
A Contribution to the Theory of Groups of Prime-Power Order .Proc. London Math. Soc. 36 (1934), s. 2, 29-95.198] R. J. Higgs.
Projective characters of degree one and the inflation restrictionsequence . J. Austral. Math. Soc. (Series A) 46 (1989), 272–280.[9] R. J. Higgs.
Subgroups of the Schur multiplier . J. Austral. Math. Soc.(Series A) 48 (1990), 497–505.[10] I. M. Isaacs.
Character Theory of Finite Groups . AMS Chelsea Publishing,American Mathematical Society, Providence, Rhode Island, 1976.[11] M. R. Jones.
Some inequalities for the multiplicator of a finite group . II,Proc. Amer. Math. Soc. 45 (1974), no. 2, 167–172.[12] G. Karpilovsky.
Group Representation, Vol. 2 . North Holland, 1993.[13] A. Lubotzky and A. Mann.
Powerfull p-groups. I. Finite groups . J. Algebra105 (1987), 484-505.[14] A. Mann.
Regular p-groups . Israel. J. Math. 10 (1971), 471-477.[15] P. Moravec.
Schur multipliers and power endomorphisms of groups . J.Algebra 308 (2007), no. 1, 12-25.[16] P. Moravec.
Schur multipliers of n-Engel groups . Internat. J. Algebra Com-put. 18 (2008), no. 6, 1101-1115.[17] P. Moravec.
On the exponent semigroups of finite p-groups . J. Group The-ory 11 (2008), 511–524.[18] P. Moravec.
On the Schur multipliers of finite p-groups of given coclass .Israel J. Math. 185 (2011), 189-205.[19] D. J. S. Robinson.
A Course in Theory of Groups . Springer-Verlag, NewYork (1987).[20] K. Tahara
On the second cohomology groups of semidirect products . Math.Zeitschrift 129 (1972), 365-379.[21] E. I. Zel’manov.
Solution of the restricted Burnside problem for groups ofodd exponent . Math. USSR Izvestiya 36 (1991), 41.[22] E. I. Zel’manov.
A solution of the restricted Burnside problem for 2-groups .Math. USSR Sb. 72 (1992), 543.[23]
Gap , Version 4.6.5 (2013), http: ////