A note on semidirect products and nonabelian tensor products of groups

Let G and H be groups which act compatibly on one another. In [2] and [8] it is considered a group construction η(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect products A⋊H into η(A,H), for finite abelian H-groups A. As a...

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Дата:	2009
Автори:	Nakaoka, I.N., Rocco, N.R.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2009
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154510
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	A note on semidirect products and nonabelian tensor products of groups / I.N. Nakaoka, N.R. Rocco // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 77–84. — Бібліогр.: 14 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1545102019-06-16T01:27:48Z A note on semidirect products and nonabelian tensor products of groups Nakaoka, I.N. Rocco, N.R. Let G and H be groups which act compatibly on one another. In [2] and [8] it is considered a group construction η(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect products A⋊H into η(A,H), for finite abelian H-groups A. As a consequence of our results we obtain that complete Frobenius groups and affine groups over finite fields are embedded into η(A,H) for convenient groups A and H. Further, on considering finite metabelian groups G in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of G. 2009 Article A note on semidirect products and nonabelian tensor products of groups / I.N. Nakaoka, N.R. Rocco // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 77–84. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20J99, 20E22 http://dspace.nbuv.gov.ua/handle/123456789/154510 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let G and H be groups which act compatibly on one another. In [2] and [8] it is considered a group construction η(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect products A⋊H into η(A,H), for finite abelian H-groups A. As a consequence of our results we obtain that complete Frobenius groups and affine groups over finite fields are embedded into η(A,H) for convenient groups A and H. Further, on considering finite metabelian groups G in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of G.
format	Article
author	Nakaoka, I.N. Rocco, N.R.
spellingShingle	Nakaoka, I.N. Rocco, N.R. A note on semidirect products and nonabelian tensor products of groups Algebra and Discrete Mathematics
author_facet	Nakaoka, I.N. Rocco, N.R.
author_sort	Nakaoka, I.N.
title	A note on semidirect products and nonabelian tensor products of groups
title_short	A note on semidirect products and nonabelian tensor products of groups
title_full	A note on semidirect products and nonabelian tensor products of groups
title_fullStr	A note on semidirect products and nonabelian tensor products of groups
title_full_unstemmed	A note on semidirect products and nonabelian tensor products of groups
title_sort	note on semidirect products and nonabelian tensor products of groups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/154510
citation_txt	A note on semidirect products and nonabelian tensor products of groups / I.N. Nakaoka, N.R. Rocco // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 77–84. — Бібліогр.: 14 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2009). pp. 77 – 84 c⃝ Journal “Algebra and Discrete Mathematics” A note on semidirect products and nonabelian tensor products of groups Irene N. Nakaoka and Noráı R. Rocco Communicated by guest editors Abstract. Let G and H be groups which act compatibly on one another. In [2] and [8] it is considered a group construction �(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect prod- ucts A⋊H into �(A,H), for finite abelian H-groups A. As a conse- quence of our results we obtain that complete Frobenius groups and affine groups over finite fields are embedded into �(A,H) for con- venient groups A and H. Further, on considering finite metabelian groups G in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of G. Dedicated to Professor Miguel Ferrero on occasion of his 70-th anniversary Introduction Let K and H be groups each of which acts upon the other (on the right), K ×H → K, (k, ℎ) 7→ kℎ; H ×K → H, (ℎ, k) 7→ ℎk and on itself by conjugation, in such a way that for all k, k1 ∈ K and ℎ, ℎ1 ∈ H, k(ℎ k1) = ( ( kk −1 1 )ℎ )k1 and ℎ(k ℎ1) = ( ( ℎℎ −1 1 )k )ℎ1 . (1) The authors acknowledge partial financial support from the Brazilian agencies CNPq (Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico) and FAPDF (Fundação de Apoio à Pesquisa do Distrito Federal). 2000 Mathematics Subject Classification: 20J99, 20E22. Key words and phrases: Semidirect products, Nonabelian tensor products, Frobenius Groups, Affine Groups. Jo u rn al A lg eb ra D is cr et e M at h .78 Nonabelian tensor products In this situation we say that K and H act compatibly on each other. An operator � in the class of (operator) groups has been introduced in [8] (see also [2] and [9]) which is defined as follows: let K, H be as above, acting compatibly on each other, and H' an extra copy of H, isomorphic through ' : H → H', ℎ 7→ ℎ', for all ℎ ∈ H. Then we define the group �(K,H) := ⟨K,H' ∣ [k, ℎ']k1 = [kk1 , (ℎk1)'], [k, ℎ']ℎ ' 1 = [kℎ1 , (ℎℎ1)'], for all k, k1 ∈ K, ℎ, ℎ1 ∈ H⟩. In particular we write �(H) for �(H,H) when all actions are conju- gations (cf. [12]). Besides its intrinsic group-theoretic interest, it follows from Proposi- tion 1.4 in [3] that there is an isomorphism from the subgroup [K,H'] of �(K,H) onto the nonabelian tensor product K ⊗H (as introduced by R. Brown and J.-L. Loday [1]), such that [k, ℎ'] 7→ k ⊗ ℎ, for all k ∈ K and ℎ ∈ H. It is worth mentioning that [K,H'] is a normal subgroup of �(K,H) and that �(K,H) = ([K,H'] ⋅K) ⋅H', where the dots denote semidirect products. On discussing nilpotency conditions on �(K,H) in [10], where K and H are nilpotent groups, we observe that even in very elementary situ- ations (in which at least one of the actions is non-nilpotent) the group �(K,H) fails to be nilpotent. In fact, with appropriate actions �(Cp, C2) contains the dihedral group of order 2p (where p denotes an odd prime), while �(V4, C3) contains the alternating group A4 (here Cn denotes the cyclic group of order n and V4 is the Klein four group; see [10] for details). In this note we are interested in embedding certain split extensions A ⋊ H into �(A,H), where A is an abelian H-group acting trivially on H. It is an easy exercise to check the compatibility of these actions for any given action of H on A. In the present situation we write �∗(A,H) for the corresponding group �(A,H). If B is any H-subgroup of A, then B ⋅H means the semidirect product of B by H. We also write [A,H] for the subgroup of A generated by the set {a−1aℎ ∣ a ∈ A, ℎ ∈ H}. With the above notation we can formulate Proposition A. If (∣A∣, ∣H∣) = 1 then [A, H] ⋅ H is embedded into �∗(A,H). If, in addition, A = [A, H] and A ∕= 1, then �∗(A,H) is non-nilpotent. In order to deal with some situations involving non-coprime actions we prove Proposition B. If A is a finite group and there is a central element ℎ ∈ H such that ℎ acts fixed-point-free (f.p.f., for short) on A, then A⋊H is embedded into �∗(A,H). In particular if F = GF (q), the finite field with q elements, then the Jo u rn al A lg eb ra D is cr et e M at h .I. N. Nakaoka, N. R. Rocco 79 affine group An(F ) is embedded into �∗(A,GLn(F )), where here A ∼= (Fn,+) is the translation subgroup. Next we shall consider finite metabelian groups G in which the derived subgroup G′ has order coprime with its index. We observe that the defining relations of �(H,K) are externalisations of commutator relations. Thus there is an epimorphism � : [G,G'] → G′, [x, y'] 7→ [x, y], for all x, y ∈ G, whose kernel we denote by J(G). As usual we write M(G) for the Schur Multiplier of G and Gab for the abelianized group G/G′. Our contribution is Proposition C. Let G be a finite metabelian group such that ∣G′∣ and ∣Gab∣ are coprime. Then (i) ∣G⊗G∣ = n∣G′∣ ⋅ ∣Gab ⊗ℤ Gab∣; (ii) ∣J(G)∣ = n∣Gab ⊗ℤ Gab∣, where n is the order of the Gab-stable subgroup of M(G′). Notation in this note is fairly standard. For elements x, y, z in an arbitrary group G, the conjugate of x by y is xy = y−1xy; the commutator of x and y is [x, y] = x−1xy and our commutators are left normed; in particular [x, y, z] = [[x, y], z]. Throughout the paper we assume that the groups K and H act com- patibly on one another. 1. Proofs Our starting point is the embedding of K ⊗ H into �(K,H) via the isomorphism K ⊗ H ∼= [K,H'] given by k ⊗ ℎ 7→ [k, ℎ'] for all k ∈ K, ℎ ∈ H (cf. [3], Proposition 1.4). By [2, Theorem 1], �(K,H) = [K,H']H'K ∼= ((K ⊗H)⋊H)⋊K. We shall use this decomposition without any further reference. This together with [1, Proposition 2.3] gives Lemma 1. The following relations hold in �(K,H) for all k, x ∈ K and ℎ, y ∈ H: (a) [k, ℎ'][x,y '] = [k, ℎ']x −1xy = [k, ℎ'](y −xy)' ; (b) [k, ℎ'] [x,y']−1 = [k, ℎ']x −yx = [k, ℎ'](y −1yx)' ; (c) [[k, ℎ'], [x, y']] = [k−1kℎ, (y−xy)']; (d) [ [k, ℎ'], [x, y']−1 ] = [k−1kℎ, (y−1yx)']. Jo u rn al A lg eb ra D is cr et e M at h .80 Nonabelian tensor products The above relations immediately lead to the Corollary 1. (a) If K acts trivially on H, then [K,H'] is abelian; (b) If K and H act trivially on each other, then [K,H'] is isomorphic to the ordinary tensor product Kab ⊗ZZ Hab of the abelianized groups. Proof of Proposition A. Since A is abelian and acts trivially on H, [5, Proposition 2.3] gives an isomorphism [A,H'] ∼= A⊗ℤHI(H), where I(H) denotes the augmentation ideal of ℤH, such that [a, ℎ'] 7→ a ⊗ (ℎ − 1). On the other hand there is an H-epimorphism � : [A,H'] → [A,H], [a, ℎ'] 7→ [a, ℎ] = a−1aℎ. It folllows from [11, 11.4.2] that Ker(�) is isomorphic to the first homology group H1(H,A). Since gcd(∣A∣, ∣H∣) = 1 we have H1(H,A) = 0 (here we use additive notation in A), so that � is an H-isomorphism. Therefore [A,H'] ∼= [A,H] and, consequently, the subgroup [A,H'] ⋅ H' of �∗(A,H) is isomorphic to the semi-direct product [A,H] ⋅ H. If in addition [A,H] = A, then certainly all terms i(� ∗(A,H)) of the lower central series of �∗(A,H) will contain the sub- group [A,H'] ∼= A. This finishes the proof. □ We recall that a finite group G containing a proper subgroup H ∕= 1 such that H ∩Hg = 1 for all g ∈ G ∖H is called a Frobenius group. The subgroup H is called a Frobenius complement. By a celebrated theorem of Frobenius, the set N = G ∖ (∪x∈G(H ∗)x) is a normal subgroup of G (called its Frobenius kernel) such that G = NH and N ∩ H = 1. We have that ∣H∣ divides ∣N ∣ − 1. If ∣H∣ = ∣N ∣ − 1, then we say that G is a complete Frobenius group; in this case the kernel N is an elementary abelian group (see for instance [14]). Corollary 2. Every finite Frobenius group with an abelian kernel A and complement H is embedded into �∗(A,H). Proof of Proposition B. Let ℎ be a central element of H such that ℎ acts f.p.f. on A. Since A is abelian and acts trivially on H, [A, ℎ'] = {[a, ℎ'] : a ∈ A} is a subgroup of �∗(A,H). Further, there is a homomorphism � : [A, ℎ'] −→ A such that [a, ℎ'] 7→ a−1aℎ. Because ℎ is central in H, we have for all a ∈ A and x ∈ H, � ([a, ℎ']x) = �([ax, ℎ']) = a−xaxℎ = a−xaℎx = ( a−1aℎ )x = (�[a, ℎ'])x . Thus � is an H-homomorphism. Further, if A = {a1, ⋅ ⋅ ⋅ , ar}, then Im(�) = {a1 −1a1 ℎ, ⋅ ⋅ ⋅ , ar −1ar ℎ}. As aℎ = a implies a = 1, it follows that ai −1ai ℎ = aj −1aj ℎ if and only if ai = aj . Hence ∣Im(�)∣ = ∣A∣. It is Jo u rn al A lg eb ra D is cr et e M at h .I. N. Nakaoka, N. R. Rocco 81 clear that ∣[A, ℎ']∣ ≤ ∣A∣. Therefore � is an H-isomorphism and A ⋊H is embedded into �∗(A,H). □ As a consequence of Proposition B we obtain Corollary 3. The affine group An(F ) is embedded into �∗(A,GLn(F )), where F denotes the finite field with q elements GF (q) and A ∼= (Fn,+) denotes the translation subgroup. Proof. Set ℎ = �In, where In denotes the identity matrix of order n and � is a generator of the multiplicative group (F∖{0}, ⋅). Then ℎ is central in GLn(F ) and acts f.p.f. on A. Thus the corollary follows from the above result. Now we observe that there is a epimorphism � : [G,G'] → G′, [x, y'] 7→ [x, y], whose kernel is denoted by J(G). Result in [1] implies that the exact sequence 1 −→ J(G) −→ [G,G'] −→ G′ −→ 1 (2) yields a central extension. On denoting by Δ(G) the subgroup ⟨[g, g']∣g ∈ G⟩ of �(G) we have that the section J(G)/Δ(G) is isomorphic to the Schur Multiplier of G (cf. [7]). We need a couple of lemmas before the proof of Proposition C. Lemma 2. ([12, Lemma 2.1] and [13, Lemma 3.1]) The following rela- tions hold in �(G), for all x, y, z ∈ G. (i) [x, y', z] = [x, y, z'] = [x, y', z']; (ii) [x', y, z] = [x', y, z'] = [x', y', z]; (iii) [g, g'] is central in �(G), for all g ∈ G; (iv) [g, g'] = 1, for all g ∈ G′; (v) If g ∈ G′ then [g, ℎ'][ℎ, g'] = 1, for all ℎ ∈ G. Lemma 3. Let G = G′ ⋅ H be a semidirect product of its subgroups G′ and H. Then in �(G), (i) [H, (G′)'] = [G′ , H']; (ii) Δ(G) = ⟨[ℎ, ℎ'] ∣ ℎ ∈ H⟩. Jo u rn al A lg eb ra D is cr et e M at h .82 Nonabelian tensor products Proof. Part (i) is a consequence of the item (v) of Lemma 2. As for part (ii), let g ∈ G be an arbitrary element. Then g = cℎ for some elements c ∈ G′ and ℎ ∈ H. Thus we have: [g, g'] = [cℎ, (cℎ)'] = [c, ℎ']ℎ[c, c']ℎ 2 [ℎ, ℎ'][ℎ, c']ℎ ' (by commutator identities) = [c, ℎ']ℎ[ℎ, ℎ'][ℎ, c']ℎ ' (by Lemma 2 (iv)) = ([c, ℎ'][ℎ, c'])ℎ ' [ℎ, ℎ'] (by Lemma 2 (iii)) = [ℎ, ℎ'], (by Lemma 2 (v)). Therefore Δ(G) = ⟨[ℎ, ℎ'] ∣ ℎ ∈ H⟩, as required. Proof of Proposition C. Firstly we observe that as gcd(∣G′∣ , ∣ ∣Gab ∣ ∣) = 1, by Schur-Zassenhaus Theorem [14, Theorem 2.7.4], there is a subgroup H of G, with H ∼= Gab, such that G = G′ ⋅H is a semidirect product of G′ and H. Further, the tensor squares H⊗H and Gab⊗Gab are isomorphic. Since Gab is abelian, Corollary 1 (b) gives Gab⊗Gab ∼= Gab⊗ZZG ab. Using Lemma 3.2 in [8] we obtain an exact sequence 1 −→ [G′ , G'] −→inc [G,G'] −→ Gab ⊗Gab −→ 1 (3) where [G,G'] ≤ �(G). As gcd(∣G′∣ , ∣ ∣Gab ∣ ∣) = 1, it follows from (2) and (3) that ∣ ∣G′ ∣ ∣ ∣ ∣ ∣ Gab ⊗ZZ Gab ∣ ∣ ∣ divides ∣[G,G']∣ . (4) On the other hand, [8, Theorem 3.3] gives that ∣[G,G']∣ divides ∣ ∣G′ ∧G′ ∣ ∣ ∣ ∣ ∣ G′ ⊗ZZ[Gab] I(G ab) ∣ ∣ ∣ ∣ ∣ ∣ Gab ⊗ZZ Gab ∣ ∣ ∣ (5) where G′∧G′ is the exterior square of the ZZ-module G′. As G′ is abelian, G′∧G′ ∼= M(G′) (cf. [7]). By [5, Proposition 5.2] we have that G′⊗ZZ[Gab] I(Gab) is isomorphic to the subgroup [G′, (Gab) ] of the group �∗(G′, Gab) (here we assume that G′ acts trivially on Gab and Gab acts on G′ induced by conjugation in G, that is, cG ′g = cg, for all c ∈ G′ and g ∈ G). Thus, Proposition A gives G′ ⊗ZZ[Gab] I(G ab) ∼= [G′, Gab] = [G′, H]. (6) Now it follows from the proof of [12, Proposition 3.5] that in �(G) [G , G'] = [G′ , (G′)'][G′ , H'][H , (G′)'][H , H'] Jo u rn al A lg eb ra D is cr et e M at h .I. N. Nakaoka, N. R. Rocco 83 where [H,H'] ∼= H ⊗H. However, by Lemma 3, [H, (G′)'] = [G′ , H'] and consequently [G , G'] = [G′ , (G′)'][G′ , H'][H , H']. (7) Since G′ and H are abelian, we have [G′, (G′)'][H,H'] ⊆ J(G), so that [G′, H] = �([G′, H']) = �([G,G']) = G′. This, together with (6), yields G′ ⊗ZZ[Gab] I(G ab) ∼= G′. (8) From (4), (5) and (8) it follows that ∣G⊗G∣ = ∣[G,G']∣ = n ∣ ∣G′ ∣ ∣ ⋅ ∣ ∣ ∣ Gab ⊗ZZ Gab ∣ ∣ ∣ (9) where n is a divisor of ∣M(G′)∣. Using (9) and sequence (2) we obtain ∣J(G)∣ = n∣Gab ⊗ℤ Gab∣. Let us show that n = ∣M(G′)H∣, where M(G′)H denotes the H-stable subgroup of M(G′) (see [6] for an overview). We observe that M(G) ∼= J(G)/Δ(G). Now by Lemma 3 (ii), Δ(G) = ⟨[ℎ, ℎ'] ∣ ℎ ∈ H⟩ ⊆ [H,H']. Considering that [H,H'] ∼= H ⊗H ∼= Gab ⊗ZZ Gab and H is abelian, we have ∣M(G)∣ = n ∣ ∣ ∣ ∣ [H,H'] ⟨[ℎ, ℎ'] ∣ ℎ ∈ H⟩ ∣ ∣ ∣ ∣ = n ∣H ∧H∣ = nM(H). (10) On the other hand, since the orders of G′ and H are coprimes, from [6, Corollary 2.2.6] M(G) ∼= M(H)× M(G′)H . (11) The required equalities then follow by (10) and (11). □ Corollary 4. Let G be a group as given in Proposition C. If M(G′) = 1, then (i) G⊗G ∼= G′ × (Gab ⊗ZZ Gab); (ii) J(G) ∼= Gab ⊗ZZ Gab. Proof. If M(G′) = 1 then previous result yields ∣J(G)∣ = ∣ ∣Gab ⊗ZZ Gab ∣ ∣ . But, according to the proof of Proposition C, J(G) contains [H,H'], which is isomorphic to Gab ⊗ZZ Gab. Hence J(G) = [H,H'] ∼= Gab ⊗ZZ Gab This proves part (ii). Part (i) follows from (ii) and the central extension (2). Jo u rn al A lg eb ra D is cr et e M at h .84 Nonabelian tensor products References [1] R. Brown, J.L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26, 1987, pp. 311-335. [2] G. Ellis, F. Leonard, Computing Schur multipliers and tensor products of finite groups, Proc. Royal Irish Acad. 95A, 1995, pp. 137-147. [3] N.D. Gilbert, P.J. Higgins, The non-abelian tensor product of groups and related constructions, Glasgow Math. J. 31, 1989, pp. 17-29. [4] D. Gorenstein, Finite Groups, Harper & Row, New York, 1968. [5] D. Guin, Cohomologie et homologie non-abelienne des groups, C. R. Acad. Sc. Paris 301, 1985, pp. 337-340. [6] G. Karpilovsky, The Schur Multiplier (London Mathematical Society mono- graphs; new ser., 2) Oxford University Press, 1987. [7] C. Miller, The second homology group of a group; relations among commutators, Proceedings AMS 3, 1952, pp. 588-595. [8] I.N. Nakaoka, Non-abelian tensor products of solvable groups, J. Group Theory 3, No 2, 2000, pp. 157-167. [9] I.N. Nakaoka, Sobre o produto tensorial não abeliano de grupos solúveis, doctoral thesis, State University of Campinas, Brazil, 1998, 85pp. [10] I.N. Nakaoka, N.R. Rocco, Nilpotent actions on non-abelian tensor products of groups, Matemática Contemporânea, 21, 2001, pp. 223-238. [11] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982. [12] N.R. Rocco, On a construction related to the non-abelian tensor square of a group, Bol. Soc. Bras. Mat. 22, 1991, pp. 63-79. [13] N.R. Rocco, A Presentation for a Crossed Embedding of Finite Solvable Groups, Comm. Algebra 22, 1994, pp. 1975-1998. [14] T. Tsuzuku, Finite Groups and Finite Geometries, Cambridge Tracts in Mathe- matics, vol. 78, Cambridge University Press, 1982. Contact information I. N. Nakaoka Departamento de Matemática Universidade Estadual de Maringá 87020-900 Maringá-PR, Brazil E-Mail: innakaoka@uem.br N. R. Rocco Departamento de Matemática Universidade de Braśılia 70910-900 Braśılia-DF, Brazil E-Mail: norai@unb.br Received by the editors: 24.08.2009 and in final form 24.09.2009.

A note on semidirect products and nonabelian tensor products of groups

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