Some structural results on the non-abelian tensor square of groups
aa r X i v : . [ m a t h . G R ] O c t Some structural results on the non-abeliantensor square of groups
Russell D. Blyth Francesco Fumagalli Marta Morigi
Abstract
We study the non-abelian tensor square G ⊗ G for the class ofgroups G that are finitely generated modulo their derived subgroup.In particular, we find conditions on G/G ′ so that G ⊗ G is isomorphicto the direct product of ∇ ( G ) and the non-abelian exterior square G ∧ G . For any group G , we characterize the non-abelian exteriorsquare G ∧ G in terms of a presentation of G . Finally, we apply ourresults to some classes of groups, such as the classes of free soluble andfree nilpotent groups of finite rank, and some classes of finite p -groups. Mathematics Subject classification.
Primary 20F05, 20F14, 20F99,20J99.
Introduction
The non-abelian tensor square G ⊗ G of a group G is a special case of thenon-abelian tensor product G ⊗ H of two arbitrary groups G and H , thatwas introduced by Brown and Loday in [6, 7] and arises from applications ofa generalized Van Kampen theorem in homotopy theory.For all g, h ∈ G let g h = ghg − and [ g, h ] = ghg − h − . Then G ⊗ G isdefined as the group generated by the symbols g ⊗ h , for g, h ∈ G , subjectto the relations gh ⊗ k = ( g h ⊗ g k )( g ⊗ k ) and g ⊗ hk = ( g ⊗ h )( h g ⊗ h k ) . The definition guarantees the existence of an epimorphism κ : G ⊗ G −→ G ′ ,defined on the generators by κ ( g ⊗ h ) = [ g, h ] for all g, h ∈ G . Let J ( G ) be1he kernel of the map κ , and let ∇ ( G ) be the normal subgroup generated bythe elements g ⊗ g , for all g ∈ G . The group ( G ⊗ G ) / ∇ ( G ) is called the non-abelian exterior square of G , and denoted by G ∧ G . The map κ factorizesmodulo ∇ ( G ), thus inducing an epimorphism κ ′ : G ∧ G −→ G ′ . By resultsin [6, 7] the kernel of the map κ ′ is isomorphic to the Schur multiplicator M ( G ) of G . Let Γ( G/G ′ ) be Whitehead’s quadratic functor, as defined in[19]. Then results in [6, 7] give a commutative diagram with exact rows andcentral extensions as columns: 1 (cid:15) (cid:15) (cid:15) (cid:15) Γ( G/G ′ ) (cid:15) (cid:15) / / J ( G ) (cid:15) (cid:15) / / M ( G ) (cid:15) (cid:15) / / / / ∇ ( G ) (cid:15) (cid:15) / / G ⊗ G κ (cid:15) (cid:15) / / G ∧ G κ ′ (cid:15) (cid:15) / / / / G ′ id / / (cid:15) (cid:15) G ′ (cid:15) (cid:15) Proposition 1
Let G be a group such that G/G ′ is finitely generated. If G/G ′ has no elements of order two or if G ′ has a complement in G then G ⊗ G ≃ ∇ ( G ) × ( G ∧ G ) . We will see that, under the hypotheses of Proposition 1, the structureof the tensor square G ⊗ G is completely determined once the structures of G/G ′ and of G ∧ G are known. In [5] Brown, Johnson and Robertson provedthat if M ( G ) is finitely generated then G ∧ G is isomorphic to the derivedsubgroup of any covering group ˆ G of G (the notion of a covering group iswell known if G is finite, see [13], and in the general case the authors of [5]adopted a similar definition).Our contribution is the following. 2 roposition 2 Let G be a group and let F be a free group such that G ≃ F/R for some normal subgroup R of F . Then G ∧ G ≃ F ′ [ F, R ] . As corollaries of Propositions 1 and 2, we deduce the structures of non-abelian tensor squares of finitely generated groups that are free in somevariety, for example, the free n -generated nilpotent groups of fixed class (seeCorollary 3.3) or the free n -generated soluble groups of fixed derived length(see Corollary 3.4).We mention here that a wide list of references on the non-abelian tensorsquare of a group G can be found in [15] and that an effective algorithm forcomputing it in the case when G is polycyclic has been recently developed byEick and Nickel in [12] and is implemented in the computing program GAP.The paper is organized as follows. In the first section we collect somebackground material and prove some new basic results on the tensor squareof an arbitrary group G . Proposition 1 is proved in Section 2, while in Sec-tion 3 we prove Proposition 2 and derive several consequences. Section 4deals with finite p -groups G ; in particular some upper bounds on the ordersof G ⊗ G and M ( G ) are found.The notation used in this paper is standard (the reader is referred forexample to [13]), with the only exception that conjugation and commutationare as defined in the second paragraph of this Introduction. Let G be an arbitrary group. In order to investigate the structure of G ⊗ G ,it is sometimes more convenient to consider the following construction, whichwas introduced in [10].Let G ϕ be a group isomorphic to G via the isomorphism ϕ : G −→ G ϕ , andconsider the group ν ( G ) := D G, G ϕ (cid:12)(cid:12) R , R ϕ , g [ g , g ϕ ] = [ g g , ( g g ) ϕ ] = g ϕ [ g , g ϕ ] , ∀ g , g , g ∈ G E , R , R ϕ are the defining relations of G and G ϕ respectively (that is, ν ( G )is the quotient of the free product G ∗ G ϕ by its normal subgroup generatedby all the words g [ g , g ϕ ] · [ g g , ( g g ) ϕ ] − and g ϕ [ g , g ϕ ] · [ g g , ( g g ) ϕ ] − , g , g , g ∈ G ). In [17] (Proposition 2.6), the non-abelian tensor square G ⊗ G is proved to be isomorphic to the commutator subgroup [ G, G ϕ ] inside ν ( G ).From now on we identify G ⊗ G with [ G, G ϕ ] and, unless differently specified,we write [ g, h ϕ ] in place of g ⊗ h (for g, h ∈ G ). For the reader’s clarity wereport here some results that we will often use. Lemma 1.1 (Lemma 2.1 in [17], Lemma 2.1 in [4])
Let G be any group.The following relations hold in ν ( G ) .(i) [ g ,g ϕ ] [ g , g ϕ ] = [ g ,g ] [ g , g ϕ ] = [ g ϕ ,g ] [ g , g ϕ ] , for all g , g , g , g ∈ G .(ii) [ g ϕ , g , g ] = [ g , g ϕ , g ] = [ g , g , g ϕ ] = [ g ϕ , g ϕ , g ] = [ g ϕ , g , g ϕ ] =[ g , g ϕ , g ϕ ] , for all g , g , g ∈ G .(iii) [ g , [ g , g ] ϕ ] = [ g , g , g ϕ ] − , for all g , g , g ∈ G .(iv) [ g, g ϕ ] is central in ν ( G ) for all g ∈ G .(v) [ g , g ϕ ][ g , g ϕ ] is central in ν ( G ) for all g , g ∈ G .(vi) [ g, g ϕ ] = 1 for all g ∈ G ′ . Corollary 1.2
Let G be any group. Then the following hold.(i) If a, b ∈ G commute, then [ a, b ϕ ] and [ b, a ϕ ] are central elements of [ G, G ϕ ] .(ii) If g ∈ G ′ or g ∈ G ′ , then [ g , g ϕ ] − = [ g , g ϕ ] .(iii) If A and B are two subgroups of G with B ≤ G ′ , then [ A, B ϕ ] = [ B, A ϕ ] .In particular, [ G, G ′ ϕ ] = [ G ′ , G ϕ ] .(iv) [ G ′ , Z ( G ) ϕ ] = 1 . Proof . (i) By Lemma 1.1 (ii) it follows that both [ a, b ϕ ] and [ b, a ϕ ] com-mute with any element of G and G ϕ , so they are indeed central elements of ν ( G ).(ii) By Lemma 1.1 (iii) the result holds if either g or g is a commutator. An4asy calculation shows that the result holds if g or g are arbitrary elementsof G ′ . (See also [18], Lemma 3.1 (iii)).(iii) Assume that A = h a i | i ∈ I i and B = h b j | j ∈ J i ≤ G ′ . Then [ A, B ϕ ] isgenerated by [ a i , b ϕj ] ( i ∈ I, j ∈ J ), which by (ii) is equal to [ b j , a ϕi ] − . Hence[ A, B ϕ ] ≤ [ B, A ϕ ], and, by a symmetric argument, we have [ A, B ϕ ] = [ B, A ϕ ].(iv) See [17], Lemma 2.7. (cid:3) These results permit a description to be provided for the derived and thelower central series of G ⊗ G in terms of those of G . Proposition 1.3
Let G be any group. Then the following hold:(i) For every n ≥ , [ G, G ϕ ] ( n ) = [ G ( n ) , ( G ( n ) ) ϕ ] .(ii) For every n ≥ , γ n +1 ([ G, G ϕ ]) = [ γ n ( G ′ ) , G ′ ϕ ] = [ G ′ , γ n ( G ′ ) ϕ ] . Proof .(i) We use induction on n . The result being trivial for n = 0, assume n = 1.For g i ∈ G ( i = 1 , . . . ,
4) we have the following identity (which is statedwithout proof in Lemma 11 of [3]):( ∗ ) [ g , g ϕ , [ g , g ϕ ]] = [ g , g ϕ ] (cid:0) [ g ,g ϕ ] [ g , g ϕ ] (cid:1) − = [ g , g ϕ ] (cid:0) [ g ,g ] [ g , g ϕ ] (cid:1) − by Lemma 1.1(i),= [ g , g ϕ , [ g , g ] ϕ ] by the defining properties of ν ( G ),= [ g , g , [ g , g ] ϕ ] by Lemma 1.1(ii).We note that both [ G, G ϕ ] ′ and [ G ′ , G ′ ϕ ] are normal in ν ( G ). By 5.1.7 of [16]we have that [ G, G ϕ ] ′ = [ G, G ϕ , [ G, G ϕ ]] is the normal closure in ν ( G ) of thesubgroup generated by the elements of the form[ g , g ϕ , [ g , g ϕ ]] , and [ G ′ , G ′ ϕ ] is the normal closure in ν ( G ) of the subgroup generated by theelements of the form [ g , g , [ g , g ] ϕ ] . Therefore ( ∗ ) shows that [ G, G ϕ ] ′ = [ G ′ , ( G ′ ) ϕ ].We now assume that the result is true for n and we prove it for n + 1.5y the inductive hypothesis and the argument above applied to the group G ( n ) , we have that[ G, G ϕ ] ( n +1) = ([ G, G ϕ ] ( n ) ) ′ = ([ G ( n ) , ( G ( n ) ) ϕ ]) ′ = [ G ( n +1) , ( G ( n +1) ) ϕ ] . (ii) The case n = 1 has already been proved in (i). Thus we assume that theresult is true for n and we prove it for n + 1. By the inductive hypothesis,we have that γ n +2 ([ G, G ϕ ]) = [ γ n +1 ([ G, G ϕ ]) , [ G, G ϕ ]] = [ γ n ( G ′ ) , G ′ ϕ , [ G, G ϕ ]] . By 5.1.7 of [16] [ γ n ( G ′ ) , G ′ ϕ , [ G, G ϕ ]] is the normal closure in ν ( G ) of thegroup generated by the elements of the form[ g , g ϕ , [ g , g ϕ ]] , with g ∈ γ n ( G ′ ) , g ∈ G ′ and g , g ∈ G. Similarly, [ γ n +1 ( G ′ ) , G ′ ϕ ] is the normal closure in ν ( G ) of the group gen-erated by the elements of the form[[ g , g ] , [ g , g ] ϕ ] , with g ∈ γ n ( G ′ ) , g ∈ G ′ and g , g ∈ G. By ( ∗ ), we have that [ g , g ϕ , [ g , g ϕ ]] = [[ g , g ] , [ g , g ] ϕ ], so that γ n +2 ([ G, G ϕ ]) =[ γ n +1 ( G ′ ) , G ′ ϕ ]. This completes the induction. Finally, by Corollary 1.2 (iii),we have [[ γ n +1 ( G ′ ) , G ′ ϕ ] = [ G ′ , γ n +1 ( G ′ ) ϕ ] , for all n ≥ (cid:3) We stress the fact that Proposition 1.3 does not say that in general ( G ⊗ G ) ( n ) and G ( n ) ⊗ G ( n ) are isomorphic groups. Consider, for example, the case G = S , where G ⊗ G is elementary abelian of order 4, while A ⊗ A has orderorder 3). Indeed, the computation of ( G ⊗ G ) ( n ) as [ G, G ϕ ] ( n ) = [ G ( n ) , ( G ( n ) ) ϕ ]occurs within the group ν ( G ), whereas the calculation of G ( n ) ⊗ G ( n ) occursas [ G ( n ) , ( G ( n ) ) ϕ ] within the group ν ( G ( n ) ).The following facts (given in [5]) are seen to be consequences of Proposition1.3 and Lemma 1.2. Corollary 1.4 If G is a solvable group of derived length d , then G ⊗ G issolvable of derived length d − or d .If G is a nilpotent group of class c , then G ⊗ G is nilpotent of class ≤ ⌊ c +12 ⌋ . The structure of the non-abelian tensor square
In this section we prove some fundamental facts about the structure of thenon-abelian tensor square of any group G such that G ab = G/G ′ is finitelygenerated. We conjecture that our results remain true under the weaker as-sumption that G/G ′ is a restricted direct product of cyclic groups.For a finitely generated abelian group A , its non-abelian tensor square issimply the ordinary tensor product of two copies of A . In particular, if A = { a , . . . , a s } is a set of generators of A such that A is the direct productof the cyclic groups h a i i , i = 1 , . . . , s , then we can write A ⊗ A = ∇ ( A ) × E A ( A ) , where ∇ ( A ) = (cid:10) [ a i , a ϕi ] , [ a i , a ϕj ][ a j , a ϕi ] | ≤ i < j ≤ s (cid:11) and E A ( A ) = (cid:10) [ a i , a ϕj ] | ≤ i < j ≤ s (cid:11) . We observe that ∇ ( A ) is independent of the set of generators A of A ,since in fact ∇ ( A ) = h [ a, a ϕ ] | a ∈ A i , while E A ( A ) is a complement of ∇ ( A )in A ⊗ A that does depend on the choice of A .It turns out that for any group G such that G ab is finitely generated (inparticular, for any finitely generated group G ), the structure of ∇ ( G ) essen-tially depends on G ab . The following Lemma, which improves Proposition3.3 of Rocco [18], makes this observation precise. Lemma 2.1
Let G be a group such that G ab is finitely generated by theelements { x i G ′ | i = 1 , . . . , s } . Set E ( G ) to be (cid:10) [ x i , x ϕj ] | i < j (cid:11) [ G ′ , G ϕ ] . Thenthe following hold:(i) ∇ ( G ) is generated by the elements of the set (cid:8) [ x i , x ϕi ] , [ x i , x ϕj ][ x j , x ϕi ] | ≤ i < j ≤ s (cid:9) .(ii) [ G, G ϕ ] = ∇ ( G ) E ( G ) . Proof .(i) Let Y = { y α } α ∈ I be a set of generators for G ′ and let X = { x i } si =1 . Then7 = X ∪ Y generates G . By Lemma 17 in [3] (or Proposition 3.3 in [18]) ∇ ( G ) is generated by { [ a, a ϕ ] , [ a, b ϕ ][ b, a ϕ ] | a, b ∈ G} . Note that [ a, a ϕ ] = 1 if a ∈ Y (by Lemma 1.1(vi)) and similarly [ a, b ϕ ][ b, a ϕ ] =1 if at least one among a and b lies in Y (Corollary 1.2 (ii)).(ii) The proof follows by a direct expansion of the factors [ x i g , ( x j g ) ϕ ]( g , g ∈ G ′ ). Alternatively, consider the map f : [ G, G ϕ ] −→ [ G ab , ( G ab ) ϕ ] in-duced by the projection onto G ab . Then Im f = f (cid:0) ∇ ( G ) (cid:10) [ x i , x ϕj ] | i < j (cid:11) (cid:1) andKer f = [ G ′ , G ϕ ] = [ G, ( G ′ ) ϕ ] (see [17], Remark 3), so [ G, G ϕ ] = ∇ ( G ) E ( G ). (cid:3) We are now able to describe the structure of the non-abelian tensor square G ⊗ G in terms of ∇ ( G ) and the non-abelian exterior square G ∧ G . Ourresult generalizes Proposition 8 in [5] and Proposition 3.1 in [4]. Proposition 2.2
Assume that G ab is finitely generated. Then, with the no-tation of Lemma 2.1, the following hold.(i) The map f defined to be the restriction f | ∇ ( G ) : ∇ ( G ) −→ ∇ ( G ab ) ofthe projection onto G ab , has kernel N = E ( G ) ∩ ∇ ( G ) . Moreover, N isa central elementary abelian -subgroup of [ G, G ϕ ] of rank at most the -rank rk ( G ab ) of G ab .(ii) [ G, G ϕ ] /N ≃ ∇ ( G ab ) × ( G ∧ G ) .(iii) Suppose either that G ab has no elements of order two or that G ′ has acomplement in G . Then ∇ ( G ) ≃ ∇ ( G ab ) and G ⊗ G ≃ ∇ ( G ) × ( G ∧ G ) . Proof .(i) Let w ∈ ∇ ( G ) ∩ E ( G ). Then f ( w ) = f ( w ) ∈ ∇ ( G ab ) ∩ E ( G ab ) = 1 , and so N ≤ Ker ( f ). Conversely, Ker ( f ) = Ker ( f ) ∩ ∇ ( G ) = ( G ′ ⊗ G ) ∩∇ ( G ) ≤ N .It is obvious that N is a central subgroup of G ⊗ G , since N is contained in ∇ ( G ). In order to show that N is an elementary abelian 2-group, we recall8hat there is a sequence of epimorphisms between finitely generated abeliangroups Γ( G ab ) ψ −→ ∇ ( G ) f −−→ ∇ ( G ab ) , where Γ( G ab ) is the Whitehead functor on G (see [5]). In particular, if N = Ker ( ψf ) and N = Ker ( ψ ), then N ≃ N /N . From [19] (II. 7), werecall some basic facts about the functor Γ. First, if A is a finitely generatedabelian group such that A = Q i A i , thenΓ( A ) = Y i Γ( A i ) × Y i Note that ∇ ( G ) /N ≃ ∇ ( G ab ) and E ( G ) /N ≃ E ( G ) ∇ ( G ) / ∇ ( G ) = ( G ⊗ G ) / ∇ ( G ) = G ∧ G. (iii) If G ab has no elements of order two, then 2 does not divide the orderof the torsion part of Γ( G ab ), and so Γ( G ab ) ≃ ∇ ( G ) ≃ ∇ ( G ab ), forcing theresult.Assume now that G ′ has a complement A in G . If we write g ∈ G as g = xa ,with x ∈ G ′ and a ∈ A , by Lemma 3.1 (iv) in [18] we have that[ g, g ϕ ] = [ a, a ϕ ] , ∇ ( G ) = h [ g, g ϕ ] | g ∈ G i = h [ a, a ϕ ] | a ∈ A i ≃ ∇ ( G ab ) , and N = 1. (cid:3) Observation. In the proof of Proposition 2.2 (i) if | x i | = | x i G ′ | , for i =1 , . . . , r , then N has rank r , so N ≃ N /N = 1, ∇ ( G ) ≃ ∇ ( G ab ) and G ⊗ G ≃ ∇ ( G ) × ( G ∧ G ). Corollary 2.3 Let G be a group such that G ab is a finitely generated abeliangroup with no elements of order two. Then J ( G ) ≃ Γ( G ab ) × M ( G ) . Proof . Note that J ( G ) is by definition the kernel of the commutator map κ : G ⊗ G −→ G ′ . In particular, J ( G ) is a central subgroup of G containing ∇ ( G ). By Proposition 2.2 we have that G ⊗ G = ∇ ( G ) × H , where H isa subgroup isomorphic to G ∧ G . Therefore, applying Dedekind’s modularlaw, we have J ( G ) = ∇ ( G ) × ( H ∩ J ( G )) ≃ Γ( G ab ) × M ( G ) , since ∇ ( G ) ≃ Γ( G ab ) and J ( G ) / ∇ ( G ) ≃ M ( G ). (cid:3) We recall the notions of non-abelian tensor center Z ⊗ ( G ) and non-abelianexterior center Z ∧ ( G ) of a group G . These groups are defined in [8] as Z ⊗ ( G ) = { g ∈ G | [ g, x ϕ ] = 1 , ∀ x ∈ G } Z ∧ ( G ) = { g ∈ G | [ g, x ϕ ] ∈ ∇ ( G ) , ∀ x ∈ G } . As Ellis showed in [8] and [9], Z ⊗ ( G ) is a characteristic central subgroup of G and is the largest normal subgroup L of G such that G ⊗ G ≃ G/L ⊗ G/L .The non-abelian exterior center Z ∧ ( G ) is a central subgroup of G and isequal to the epicenter Z ∗ ( G ) of G .. In particular, a group G is capable (thatis, is isomorphic to a central quotient E/Z ( E ) for some group E ) if and onlyif Z ∧ ( G ) = 1. Corollary 2.4 Let G be any group such that G ab is finitely generated. Withthe notation of Proposition 2.2, if N = 1 then Z ⊗ ( G ) = Z ∧ ( G ) ∩ G ′ . Inparticular, the conclusion holds if G is a finite group of odd order. roof . By the definition of exterior center we have that[ Z ∧ ( G ) ∩ G ′ , G ϕ ] ≤ N = 1 . Therefore Z ∧ ( G ) ∩ G ′ ≤ Z ⊗ ( G ). Conversely, we trivially have Z ⊗ ( G ) ≤ Z ∧ ( G ). Moreover, Z ⊗ ( G ) ≤ G ′ , as if x ∈ Z ⊗ ( G ) then [ G ′ x, ( G ′ x ) ϕ ] shouldbe the trivial element of the tensor product G ab ⊗ G ab , being the image of1 = [ x, x ϕ ] under the natural map from G ⊗ G to G ab ⊗ G ab . This of courseforces G ′ x being the identity element of G ab , so x ∈ G ′ . (cid:3) Question 1 With the notation of Proposition 2.2, is it always true that N = [ Z ∧ ( G ) ∩ G ′ , G ϕ ]?Note that a positive answer to the previous question will imply, by Proposi-tion 9 in [5], that G ⊗ GN ≃ GH ⊗ GH , where H is defined to be Z ∧ ( G ) ∩ G ′ . This is precisely the case for generalizedquaternion groups and semi-dihedral groups, as the reader may check bysome easy calculations and the aid of [5], Proposition 13, and [8], Proposition16. We will now describe the structure of the non-abelian exterior square G ∧ G of a group G . Throughout this section we view the non-abelian tensor square G ⊗ G as defined at the beginning of the paper, with generators g ⊗ g , ratherthan via the isomorphic subgroup [ G, G ϕ ] of ν ( G ). We denote with g ∧ g the coset of G ∧ G containing g ⊗ g .Let G be a group and let R i −→ F π −→ G be a presentation for G , where F isa free group. Set F ◦ to be the quotient F/ [ F, R ] and set R ◦ to be R/ [ F, R ],so that 1 −→ R ◦ i −→ F ◦ η −→ G −→ ξ : G ⊗ G −→ ( F ◦ ) ′ η ξ is the commutator map κ : G ⊗ G −→ G ′ . In particular, ξ operates as follows on the generators g ⊗ g of G ⊗ G : ξ ( g ⊗ g ) = [ f , f ][ F, R ] , where f and f are any two preimages of g and g in F , respectively. Ofcourse, ξ is trivial on the central subgroup ∇ ( G ), and so it induces a homo-morphism ξ : G ∧ G −→ ( F ◦ ) ′ . (2)The following Proposition is the main result of this section. The proof usesan argument similar to that of Theorem 2 in [14]. Proposition 3.1 Let G be a group and let F be a free group such that G ≃ F/R for some normal subgroup R of F . Then G ∧ G ≃ F ′ [ F, R ] . Proof . We will show that the map ξ defined in (2) is an isomorphism.The surjectivity of ξ is immediate. Let π : F → G be the projection withkernel R . An arbitrary generator [ f , f ][ F, R ] of F ′ / [ F, R ] lies in the imageof ξ , since ξ ( π ( f ) ∧ π ( f )) = [ f , f ][ F, R ].We now prove that ξ is injective. Using the same notation as in the in-troduction, for any group X we set J ( X ) to be Ker ( κ ) and M ( X ) to beKer ( κ ′ ), where κ and κ ′ are the commutator maps κ : X ⊗ X −→ X ′ and κ ′ : X ∧ X −→ X ′ respectively, so that J ( X ) / ∇ ( X ) ≃ M ( X ).Let φ be the map ξ restricted to M ( G ). We want to show that φ is injective.Note that φ is a map φ : M ( G ) −→ R ◦ ∩ ( F ◦ ) ′ = F ′ ∩ R [ F, R ] . Now the quotient map η : F ◦ −→ G induces a homomorphism η ∗ : M ( F ◦ ) −→ M ( G )(which is the restriction to M ( F ◦ ) of the map sending f [ F, R ] ∧ f [ F, R ] to η ( f ) ∧ η ( f )). It is easy to notice that the following is an exact sequence M ( F ◦ ) η ∗ −→ M ( G ) φ −→ R ◦ ∩ ( F ◦ ) ′ . φ is injective, we will prove that η ∗ is thetrivial map. If α : J ( F ◦ ) −→ M ( F ◦ ) is the quotient map, we show that η ∗ ( α ( J ( F ◦ )) = η ∗ ( α ( ∇ ( F ◦ )).Let w = Q ( x i ⊗ y i ) ∈ J ( F ◦ ), with x i , y i ∈ F ◦ . Then there exist x i , y i ∈ F and w = Q ( x i ⊗ y i ) ∈ F ⊗ F such that λ ⊗ ( w ) = w and λ ( κ ( w )) = κ ( w ) = 1 (here λ ⊗ : F ⊗ F −→ F ◦ ⊗ F ◦ is the map induced by the projection λ : F −→ F ◦ ).Thus κ ( w ) = Q [ x i , y i ] ∈ Ker ( λ ) = [ F, R ]. As F is a free group its Schurmultiplicator is trivial, so M ( F ) = 1, that is, J ( F ) = ∇ ( F ). In particular,modulo ∇ ( F ), the product w is equivalent to Q ( f j ⊗ r j ), for some f j ∈ F and r j ∈ R . So w = Q ( f j ⊗ r j ) z , for some z ∈ ∇ ( F ). We note that ( α ( λ ⊗ ( z )) = 1.Now η ∗ ( α ( w )) = η ∗ ( α ( λ ⊗ ( w ))) = η ∗ ( α ( λ ⊗ ( Y ( f j ⊗ r j ) z )))= Y ( η ∗ α ( f j ⊗ , therefore η ∗ is the trivial map and φ is injective.Finally in order to show that ξ is an isomorphism between G ∧ G and F ′ / [ F, R ], we apply the Short Five Lemma ([1], Proposition 2.10) to thefollowing commutative diagram.1 −−−→ M ( G ) i −−−→ G ∧ G κ ′ −−−→ G ′ −−−→ y φ y ξ y G ′ −−−→ R ∩ F ′ [ F,R ] i −−−→ F ′ [ F,R ] η −−−→ G −−−→ φ and the identity map of G ′ are injective, it follows that also ξ isinjective. This concludes the proof that ξ is an isomorphism. (cid:3) As consequences of the results above we now describe the structures of thenon-abelian tensor squares of some groups that are “universal” in the sensethat they are free in suitable varieties. Corollary 3.2 ([5], Proposition 6) Let F n be a free group of rank n . Then F n ⊗ F n ≃ Z n ( n +1) / × ( F n ) ′ . Proof . Since F abn is a free abelian group of rank n , by Proposition 2.2 wehave ∇ ( F n ) ≃ ∇ ( F abn ) ≃ Z n ( n +1) / and F n ⊗ F n ≃ ∇ ( F abn ) × F n ∧ F n . FinallyProposition 3.1 (with F = F n and R = 1) gives F n ∧ F n ≃ ( F n ) ′ . (cid:3) Corollary 3.3 ([4], Corollary 1.7) Let G = N n,c be the free nilpotent groupof rank n > and class c ≥ . Then G ⊗ G ≃ Z n ( n +1) / × ( N n,c +1 ) ′ . Proof . As before, G ab is a free abelian group of rank n , and so we have ∇ ( G ) ≃ ∇ ( G ab ) × Z n ( n +1) / and G ⊗ G ≃ ∇ ( G ab ) × G ∧ G . Finally applyProposition 3.1 with F free group of rank n and R = γ c +1 ( F ). (cid:3) Corollary 3.4 Let F be the free group of finite rank n > , let d be a naturalnumber, and let G = F/F ( d ) be the free solvable group S n,d of derived length d and rank n > . Then G ⊗ G ≃ Z n ( n +1) / × F ′ / [ F, F ( d ) ] is an extension of a nilpotent group of class ≤ by a free solvable group ofderived length d − and infinite rank. In particular, if d = 2 , then G ⊗ G isa nilpotent group. Proof . Once again, Propositions 2.2 and 3.1 imply that G ⊗ G has thedescribed factorization. Note that F ( d − / [ F, F ( d ) ] is a normal subgroup ofthe group F ′ / [ F, F ( d ) ] and that F ( d − / [ F, F ( d ) ] is nilpotent of class at most 3,as it is a quotient of F ( d − /γ ( F ( d − ). So M = Z n ( n +1) / × F ( d − / [ F, F ( d ) ] isalso nilpotent of class at most 3 and G ⊗ G/M is isomorphic to F ′ /F ( d − , soit is free solvable of derived length d − 2. The fact that F ′ /F ( d − is of infiniterank follows from the well-known fact that F ′ is not finitely generated. (cid:3) We recall, in view of Theorem A in [2], that the Schur multiplicator of S n,d is not finitely generated. In particular, S n,d ∧ S n,d can also be viewed as anextension of a central abelian group of infinite rank by a free solvable groupof derived length d − p -group.Let d be an integer and, as before, denote by F d the free group on d gen-erators. We recall that for every integer i the group γ i ( F d ) /γ i +1 ( F d ) is free14belian of rank m d ( i ) := 1 i X t | i µ ( t ) d i/t , where µ is the Mobius function (see [13] Chapter 3.2).We also recall for a fixed prime number p the notion of lower central p -series of a group G . The terms of this series are { λ i ( G ) } i ≥ , where λ ( G ) = Gλ k +1 ( G ) = [ λ k ( G ) , G ] λ k ( G ) p , for k ≥ . We note that this series is the most rapidly descending central series of G whose factors have exponent p (see [13], Chapter 3). The lower central p -series will be used in the next section to find some bounds on the ordersof the non-abelian tensor and exterior squares of finite p -groups. Now weexhibit an explicit calculation of these objects in a particular case.For every pair of positive integers d and c define G d,c to be the quotient F d /λ c +1 ( F d ). According to [13] (Theorem 3.2.10), G d,c is a finite p -group ofclass c and order p m , where m = P cj =1 ( c + 1 − j ) m d ( j ). Corollary 3.5 With the above notation, we have that G d,c ∧ G d,c ≃ ( G d,c +1 ) ′ and G d,c ⊗ G d,c ≃ ( Z p c ) d ( d +1) / × ( G d,c +1 ) ′ . Proof . Let G := G d,c . We first prove that G ⊗ G ≃ ∇ ( G ) × ( G ∧ G ) . (3)For p odd (3) follows from Proposition 2.2, while for the case p = 2 a littlemore care is needed. More precisely, we observe that if F d = h f , . . . , f d i ,then the image x i in G = F d /λ c +1 ( F d ) of the generator f i of F d has order p c for each i = 1 , . . . , d . Moreover, by Theorem 3.2.10 in [13], G ab is isomorphicto a direct product of d = m d (1) cyclic groups Z p c of order p c . So now ourresult follows from the observation following Proposition 2.2.We have that ∇ ( G ) ≃ ( Z p c ) d ( d +1) / . We claim that the derived subgroupof a covering group for G is isomorphic to ( G d,c +1 ) ′ . In the following, let L i denote λ i ( F d ), i ≥ 1. We note that the group G d,c +1 = F d /L c +2 has L c +1 /L c +2 as a central elementary abelian subgroup. Moreover, the subgroup ML c +2 defined to be ( G d,c +1 ) ′ ∩ L c +1 L c +2 = γ ( F d ) L c +2 ∩ L c +1 L c +2 15s isomorphic to L c +1 ∩ γ ( F d ) L c +2 ∩ γ ( F d ) , which is isomorphic to M ( G ), by Theorem 3.2.10 in [13]. Now let H/L c +2 be a complement of M/L c +2 in L c +1 /L c +2 and consider the factor group G d,c +1 = G d,c +1 H/L c +2 . If N ≤ G d,c +1 we denote with N the image of N in G d,c +1 under the canonicalprojection; it follows that M ( G ) ≃ M ≤ Z ( G d,c +1 ) ∩ ( G d,c +1 ) ′ . Moreover, G d,c +1 /M ≃ F d /L c +1 = G , so G d,c +1 is a covering group for G .Finally, note that( G d,c +1 ) ′ = ( F d ) ′ HH ≃ ( F d ) ′ ( F d ) ′ ∩ H = ( F d ) ′ ( F d ) ′ ∩ L c +2 = ( G d,c +1 ) ′ . (cid:3) p -groups Throughout this section G is a finite p -group, for some prime p . We startwith a lemma concerning the lower central p -series of G . We again identifythe group G ⊗ G with its isomorphic image [ G, G ϕ ] in the group ν ( G ) definedin Section 2. Lemma 4.1 Let G be a finite p -group. Then for every k ≥ , [ λ k ( G ) , G ϕ ] = [ G, ( λ k ( G )) ϕ ] . Proof . We prove the result by induction on k . Since the result istrivial for k = 1, we assume [ λ k ( G ) , G ϕ ] = [ G, ( λ k ( G )) ϕ ] and show that[ λ k +1 ( G ) , G ϕ ] = [ G, ( λ k +1 ( G )) ϕ ].First note that, since [ λ k ( G ) , G, G ϕ ] and [ λ k ( G ) p , G ϕ ] are both normal in16 ( G ), we have, using [ xy, a ϕ ] = x [ y, a ϕ ][ x, a ϕ ] with x ∈ [ λ k ( G ) , G ], y ∈ λ k ( G ) p and a ∈ G , that [ λ k +1 ( G ) , G ϕ ] = [ λ k ( G ) , G, G ϕ ][ λ k ( G ) p , G ϕ ] . (As [ xy, a ϕ ] = x [ y, a ϕ ][ x, a ϕ ], with x ∈ [ λ k ( G ) , G ], y ∈ λ k ( G ) p and a ∈ G ).Using Lemma 1.1(ii), we have that[ λ k ( G ) , G, G ϕ ] = [ λ k ( G ) ϕ , G ϕ , G ] = [ G, [ λ k ( G ) , G ] ϕ ] ≤ [ G, λ k +1 ( G ) ϕ ] . Thus our proof will be complete if we show that [ λ k ( G ) p , G ϕ ] ≤ [ G, λ k +1 ( G ) ϕ ].Define R to be [ λ k ( G ) , G, G ϕ ] (= [ G, [ λ k ( G ) , G ] ϕ ]).Note that R contains the derived subgroup of [ λ k ( G ) , G ϕ ]. To see this, weobserve that [ λ k ( G ) , G ϕ ] ′ is generated by the elements[[ x, a ϕ ] , [ y, b ϕ ]] , where x, y ∈ λ k ( G ) and a ϕ , b ϕ ∈ G ϕ , and, by Lemma 1.1(i) and the defining properties of ν ( G ), we have that[[ x, a ϕ ] , [ y, b ϕ ]] = [[ x, a ] , [ y, b ] ϕ ] ∈ R .We claim that the following hold:[ x m , a ϕ ] ∈ [ x, a ϕ ] m R for all x ∈ λ k ( G ) , a ϕ ∈ G ϕ , m ∈ N (4)[ y, ( b m ) ϕ ] ∈ [ y, b ϕ ] m R for all y ∈ G, b ϕ ∈ ( λ k ( G )) ϕ , m ∈ N . (5)We prove (4) by induction on m . The proof of (5) is similar.If m = 1 then (4) is trivially true. Let m ≥ 2. Then[ x m , a ϕ ] = [ x · x m − , a ϕ ] = x [ x m − , a ϕ ][ x, a ϕ ]= [ x m − , ( x a ) ϕ ][ x, a ϕ ] . Now the claim is proved since, by induction on m , the term [ x m − , ( x a ) ϕ ] liesin the coset[ x, ( x a ) ϕ ] m − R = [ x, [ x, a ] ϕ a ϕ ] m − R == ([ x, [ x, a ] ϕ ] · [ x,a ] ϕ [ x, a ϕ ]) m − R == ([ x, [ x, a ] ϕ ] · [ x, a ϕ ]) m − R = ([ x, a ϕ ]) m − R, by a repeated use of Lemma 1.1, and the fact that [ x, [ x, a ] ϕ ] = [ x, a, x ϕ ] − ∈ R . Therefore our claims (4) and (5) are true, and we now complete the17roof of the lemma as follows. We have that [ λ k ( G ) p , G ϕ ] is generatedby elements of the form [ x p , a ϕ ] with x ∈ λ k ( G ) and a ϕ ∈ G ϕ . By (4),[ x p , a ϕ ] ∈ ([ x, a ϕ ]) p R . Now [ x, a ϕ ] ∈ [ λ k ( G ) , G ϕ ] = [ G, ( λ k ( G )) ϕ ] by theinductive hypothesis, so we may write[ x, a ϕ ] = w · . . . · w l , where w i = [ y i , b ϕi ], y i ∈ G and b ϕi ∈ λ k ( G ) ϕ for i = 1 , . . . , l . In particular,since [ λ k ( G ) , G ϕ ] /R is abelian we have that ([ x, a ϕ ]) p R = w p . . . w pl R. Finally,by (5) w pi R = [ y i , ( b pi ) ϕ ] R for all i = 1 , . . . , l , forcing[ x p , a ϕ ] ∈ R [ G, ( λ k ( G ) p ) ϕ ] = [ G, ( λ k +1 ( G )) ϕ ] . (cid:3) The following result is an improvement of Corollary 3.12 in [17]. In his PhDthesis A. McDermott proves this result using arguments different from ours(see [11]). Proposition 4.2 Let G be a finite group of order p n ( p a prime) and let d = d ( G ) be the minimum number of generators of G . Then p d ≤ | [ G, G ϕ ] | ≤ p nd . Proof . Of course | [ G, G ϕ ] | ≥ p d , as G ⊗ G admits G/ Φ( G ) ⊗ G/ Φ( G ) asa quotient, and G/ Φ( G ) ⊗ G/ Φ( G ) is elementary abelian of order p d , sinceit is an ordinary tensor product.Let λ k ( G ) be the last non-trivial term of the series { λ i ( G ) } i , and let π : G −→ G := G/λ k ( G ) be the quotient map. π induces a natural epimorphism, say, e π : [ G, G ϕ ] −→ [ G, G ϕ ]. According to [17, Remark 3] and using the previousLemma, the kernel of e π consists in the subgroupKer ( e π ) = [ λ k ( G ) , G ϕ ][ G, λ k ( G ) ϕ ] = [ λ k ( G ) , G ϕ ] . Since λ k ( G ) is a central elementary abelian subgroup of G , by Lemma 1.1(ii),we have that Ker ( e π ) is an elementary abelian p -subgroup lying in the centerof ν ( G ). Thus the map θ : λ k ( G ) × G −→ [ λ k ( G ) , G ϕ ]( a, g ) [ a, g ϕ ] , 18s bilinear. Let λ k ( G ) be generated by the set { a i | i = 1 , . . . , d k } and let G begenerated by { g i | i = 1 , . . . , d } . Therefore Ker ( e π ) is generated by the set (cid:8) [ a i , g ϕj ] | i = 1 , . . . , d k , j = 1 , . . . , d (cid:9) , forcing | Ker ( e π ) | ≤ p d · d k , and | [ G, G ϕ ] | ≤ p d · d k (cid:12)(cid:12) [ G, G ϕ ] (cid:12)(cid:12) . By induction weobtain that | [ G, G ϕ ] | ≤ p d · d k · . . . · p d = p d P ki =1 d i = p nd . (cid:3) Remark 1 Homocyclic abelian groups show that the upper bound in theProposition 4.2 is best possible. Another example in which the upper boundis reached is when G is the group G , := F /λ ( F ) . As a consequence of our results we have the following bound for the order ofthe Schur multiplicator of finite p -groups. Corollary 4.3 Let G be a finite p -group of order p n with d = d ( G ) genera-tors. If p is odd, the order of the Schur multiplicator M ( G ) of G is at most p d ( n − ( d +1) / . If p = 2 , then | M ( G ) | ≤ d ( n − ( d +3) / . Proof . By Proposition 3.1 and the definition of the exterior square | M ( G ) | | G ′ | = | G ∧ G | = | G ⊗ G ||∇ ( G ) | . If p is odd, by Proposition 2.2, ∇ ( G ) ≃ ∇ ( G ab ), and so |∇ ( G ) | ≥ p d ( d +1) / .If p = 2, then |∇ ( G ) | ≥ p d ( d +3) / . The proof is now completed by using thebounds given in Proposition 4.2. (cid:3) References [1] M.F. Atiyah, I.G. 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