A note on the invariance in the nonabelian tensor product

A NOTE ON THE INV ARIANCE IN THE NONABELIAN TENSOR PR ODUCT FRANCESCO G. RUSSO Abstract. In the nonabelian tensor pro duct G ⊗ H of tw o groups G and H man y prop erties pass from G and H to G ⊗ H . There is a wide literature for differen t prop erties i n v olve d i n this passage. W e l ook at wea k conditions for which suc h a passage ma y happen. 1. Terminology and st a tement of the resul t Let G and H b e t w o g r oups acting upon each other in a compatibl e w ay : (1.1) g h g ′ = g ( h ( g − 1 h ′ )) , h g h ′ = h ( g ( h − 1 h ′ )) , for g , g ′ ∈ G and h, h ′ ∈ H , and acting upon themse lves by conjugation. The nonabel ian te nsor produc t G ⊗ H of G and H is the group genera ted by the symbols g ⊗ h with defining relatio ns (1.2) g g ′ ⊗ h = ( g g ′ ⊗ g h )( g ⊗ h ) , g ⊗ hh ′ = ( g ⊗ h )( h g ⊗ h h ′ ) . When G = H and all actions ar e b y conjugations, G ⊗ G is ca lled nonabel ian tensor sq uar e of G . These notions were introduced in [3, 4] and some s ignificant contributions ca n b e found in [1, 2, 5, 6, 8, 9, 10, 1 2, 13]. F rom the defining r elations in G ⊗ H , (1.3) κ : g ⊗ h ∈ G ⊗ H 7→ κ ( g ⊗ h ) = [ g , h ] ∈ [ G, H ] = h g − 1 h − 1 g h | g ∈ G, h ∈ H i is an epimo rphism of groups. Still from [3, 4], if G and H act trivially up on each other, then G ⊗ H is isomo rphic to the usual tenso r pro duct G ab ⊗ Z H ab . If they act compatibly up on each other, then their a ctions induce an a ction o f the free pro duct G ∗ H on G ⊗ H g iv en by x ( g ⊗ h ) = x g ⊗ x h , where x ∈ G ∗ H . The exter ior produ ct G ∧ H is the g roup obtained with the additional relation g ⊗ h = 1 ⊗ on G ⊗ H , that is, (1.4) G ∧ H = ( G ⊗ H ) /D , where D = h g ⊗ g : g ∈ G ∩ H i . Now it is eas y to c heck that (1.5) κ ′ : g ∧ h ∈ G ∧ H 7→ κ ′ ( g ∧ h ) = [ g , h ] ∈ [ G, H ] is a well–defined epimor phism of gro ups. F or conv enience of the r e a der, we reca ll that there is a famous commutativ e diagram with exact r ows and central extensions as columns in [3 , (1)]: It correla tes the se c o nd ho mology group H 2 ( G ) of G with the third ho mology group H 3 ( G ) of G , the Whitehead’s quadratic functor Γ, the Whitehead’s function ψ and ker κ = J 2 ( G ) (see also [3, 4 , 14]). Date : Nov ember 13, 2018. Key wor ds and phr ases. Nonabelian tensor pro duct; classes of groups; univ ersal property . Mathematics Subje ct Classific ati on 2010 : P r imary 20J99; Secondary 20F18. 1 2 FRANCESCO G. RUSSO Now w e get to the purp ose o f the present pape r . Given a class of gro ups X , many authors ans w ered the question: (1.6) If G, H ∈ X , then G ⊗ H ∈ X In case X = F is the cla s s of all finite g roups, s ee [5]. In ca se X = N is the class of all nilp otent groups , s ee [2, 13]. In ca se X = S is the clas s of a ll s oluble gr oups, see [10, 13]. In case X = P is the cla ss o f a ll p olycyclic gro ups, see [8]. In case X = L F is the class of a ll lo cally finite gro ups , s ee [9]. In case X = ˇ C (resp., X = S 2 ) is the class of all Cherniko v (res p., soluble minimax) gro ups, see [11]. Some top olog ical prop erties are also closed with res pect to forming the no nabe lia n tenso r pro duct, as o bserved in [3, 4]. W e recall some no tations fro m [7]. – X = S X mea ns that X is close d with r esp ect to forming subgroups. – X = H X mea ns that X is closed with resp ect to forming homomor phic images. – X = P X means that X is clos ed with resp ect to forming extensio ns, i.e.: if N ∈ X is a normal subgroup o f G and G/ N ∈ X , then G ∈ X . – X = H 2 X means that X is closed with resp ect to forming the second ho - mology group, i.e.: if G ∈ X , then H 2 ( G ) ∈ X . – X = H 3 X mea ns that X is closed with res p ect to forming the third ho mo logy group, i.e.: if G ∈ X , then H 3 ( G ) ∈ X . – X = T X means that X is closed with respect to forming (usual) abelian tensor pro ducts , i.e.: if A, B ∈ X a re ab elian, then A ⊗ Z B ∈ X . Our main contribution is below. Main T heorem. L et G and H b e two gr oups, acting c omp atibly up on e ach other and X = S X = H X = P X = H 2 X = H 3 X = T X . If G, H , Γ(( G ∩ H ) ab ) ∈ X , then G ⊗ H ∈ X . In [2, 5, 8, 9 , 10, 11, 13], the quoted results follow fr om Main The o rem, when we choose X among F , N , S , P , L F , ˇ C , S 2 . 2. Proof and s ome consequences W e illustrate that it is po ssible to adapt an ar gumen t in [8, Sectio n 2]. Pr o of of Main The or em. Let P = G ∗ H /I J b e the Pfeiffer pro duct of G and H , where I and J are the nor ma l closure s in G ∗ H of h h g hg − 1 h − 1 : g ∈ G, h ∈ H i and h g hg h − 1 g − 1 : g ∈ G, h ∈ H i , r esp ectiv ely . See [8 , 14]. Note that P is a homomorphic image of G ⋉ H , henc e P ∈ X . Here we hav e used X = H X . Let µ : G → P and ν : H → P b e inclusions. Denote G = µ ( G ) and H = ν ( H ). Then G and H are nor mal subg r oups of P and P = G H . O f co urse, ker µ ≤ Z ( G ) and ker ν ≤ Z ( H ). An a rgument as in [3, P r op osition 9] shows that the following sequence is exact: (2.1) ( G ⊗ k er ν ) × (k er µ ⊗ H ) i − → G ⊗ H − → G ⊗ H − → 1 , where i is the inclusio n ( g ⊗ h ′ , g ′ ⊗ h ) 7→ ( g ⊗ h ′ )( g ′ ⊗ h ). It is easy to see that Im i ≤ Z ( G ⊗ H ). Since h g = ν ( g ) g and g h = µ ( g ) h , ker µ and ker ν act tr ivially o n H and G , resp ectively . Therefore, (2.2) G ⊗ ker ν ≃ G ab ⊗ Z ker ν ab = G ab ⊗ Z ker ν A NOTE ON THE INV ARIANCE IN THE NONABELIAN TENSOR PR ODUCT 3 and (2.3) ker µ ⊗ H ≃ ker µ ab ⊗ Z H ab = ker µ ⊗ Z H ab . In particular, G ⊗ ker ν ∈ X . Here we hav e used X = T X . Analogously , ker µ ⊗ H ∈ X . It follows that Im i ∈ X be c ause it is a homo morphic image of ( G ⊗ ker ν ) × (ker µ ⊗ H ) ∈ X . Still we hav e us ed X = H X . Since G ⊗ H ≃ ( G ⊗ H ) / Im i , it is enough to prov e that G ⊗ H ∈ X . Here we hav e used X = P X . W e may work with G instead o f G a nd with H instead of H in order to get our result. Then ther e is no loss of gener ality in assuming that G and H ar e nor mal subg roups of P , P = GH , and all actio ns ar e induced by conjuga tion in P . Note that ( G ∧ H ) / ker κ ′ is isomorphic to [ G, H ] ≤ G ∩ H ≤ G ∈ X and so ( G ∧ H ) / ker κ ′ ∈ X . Her e we hav e used X = H X = S X . If we pr ov e ker κ ′ ∈ X , then G ∧ H ∈ X by X = P X . If we prove also D ∈ X , then G ⊗ H ∈ X , still by X = P X and we are done. By [4, Theor em 4.5], we ha ve an exact sequence: (2.4) − → H 3 ( P /G ) ⊕ H 3 ( P /H ) − → ker κ ′ − → H 2 ( P ) − → . Since P , P /G, P /H ∈ X , we have H 2 ( P ) , H 3 ( P /G ) , H 3 ( P /H ) ∈ X . Here we hav e used X = H X = H 2 X = H 3 X . On the o ther hand, k er κ ′ is an extension of H 3 ( G/ M ) ⊕ H 3 ( G/ N ) ∈ X b y H 2 ( G ) ∈ X . Therefore , ker κ ′ ∈ X , as claimed. Here we have used X = P X . Having in mind the famous diagr am [3, (1 )], it is easy to chec k that there exists a well–defined homomorphism o f groups ψ : Γ(( G ∩ H ) ab ) → ( G ∩ H ) ⊗ ( G ∩ H ). See [3, p.181] or [4]. Then Im ψ = D ∈ X , a s claimed. Here we hav e used X = H X and Γ(( G ∩ H ) ab ) ∈ X . The result follows.  Note that Γ( G ab ) plays a fundamen tal ro le in deciding if G ⊗ G ∈ X . This was alr eady no ted in [2, Section 3] for the clas s of all free nilpo ten t g roups of finite rank. Then it is clea r that the following cor ollary extends many results in [2, 5, 8, 9 , 1 0, 11, 13] in case of the nonab elian tenso r squar e. Corollary . Assume G = H in Main The or em. If G, Γ( G ab ) ∈ X , then G ⊗ G ∈ X . W e end with tw o observ ations o n the inv ariance with re spect to the nonabelia n tensor pro duct. Remark 1. Sometimes it is enough that [ G, H ] ∈ X in order to decide whether G ⊗ H ∈ X . In case of X = N , or X = S , this can be found in [3, 10, 13]. The second dea ls with the un iversal pr op erty o f the nonab elian tensor pro ducts. Remark 2. In a certa in s e ns e the universal pr op erty of the nonab elian t ensor pr o ducts (see [4]) justifies Ma in Theorem, b e cause it s hows that we need at least X = S X = H X = P X , if we hop e to a nswer (1.6) pos itiv ely . References [1] M. Bac on, L.-C. Kapp e and R. F. M orse, On the nona bel i an ten sor square of a 2-Engel group, Ar ch. Math. 69 (1997), 353–364. 4 FRANCESCO G. RUSSO [2] R. D. Blyth, P . Morav ec and R. F. Morse, On the nonab elian tensor squares of free nilp oten t groups of finite rank, In: Computational gr oup t he ory and the the ory of gr oups , Conte mp. Math., 470, Am er. M ath. So c., Providence , RI, 2008, pp. 27–43. [3] R. Brown, D. L. Johnson and E. F. Robertson, Some computations of non-ab elian tensor products of gr oups, J. Algebr a 1 11 (1987), 177–202. [4] R. Brown and J. -L. Lo da y , V an Kamp en theorems for diagrams of spaces, T op olo gy 26 (1987), 311–335. [5] G. El lis, The nonab elian tensor pro duct of finite groups i s finite, J. 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V i ssc her, On the nilp otency class and solv abil it y length of nonab elian tensor product of groups, Ar ch. Math. (Basel) 73 (1999), 161–171. [14] J. H. C. Whitehead, A c ertain ex act se quenc e . Ann. of Math. 52 (1950), 51–110. Curr ent addr ess : Lab oratorio di Dinamica Strutturale e Geotecnica (StreGa ), Univ ersit´ a del Molise, via D uca degli Abr uzzi, 86039, T ermoli (CB). E-mail addr ess : francescog.rus so@yahoo.com