View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 276 (2004) 193–209 www.elsevier.com/locate/jalgebra On the structure of normal subgroups of potent p-groups J. González-Sánchez a,1 and A. Jaikin-Zapirain b,∗,2 a Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Spain b Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Spain Received 18 December 2002 Communicated by Leonard L. Scott, Jr. Abstract 4 p Let G be a finite p-group satisfying [G, G] G for p = 2andγp−1(G) G for p>2. The main goal of this paper is to show that any normal subgroup N of G lying in G2 is power pk pk k abelian, that is, the following holds: (1) N ={g | g ∈ N};(2)Ωk(N) ={g ∈ N | o(g) p }, pk and (3) |N |=|N : Ωk(N)|. 2004 Elsevier Inc. All rights reserved. 1. Introduction Let G be a finite p-group. Consider the following subgroups of G: pi pi pi G = g | g ∈ G ,Ωi(G) = g ∈ G | g = 1 . If G is abelian, we have the following simple description of these groups: i i (1) Gp ={gp | g ∈ G}, i (2) Ωi(G) ={g ∈ G | o(g) p }. * Corresponding author. E-mail addresses: [email protected] (J. González-Sánchez), [email protected] (A. Jaikin-Zapirain). 1 The work of author has been partially supported by the Government of the Basque Country. 2 The work of author has been partially supported by the MCYT Grants BFM2001-0201, BFM2001-0180, FEDER and the Ramón y Cajal Program. 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2003.12.006 194 J. González-Sánchez, A. Jaikin-Zapirain / Journal of Algebra 276 (2004) 193–209 Moreover, the next equalities hold pi (3) |G |=|G : Ωi(G)| for all i. In the following we call a finite p-group G power abelian if it satisfies these three properties for all i. It is very interesting problem to determine which groups are power abelian. In [4] P. Hall introduced the notion of regular groups and showed that they are power abelian. The notion of powerful p-groups was introduced by A. Mann and after the work of A. Lubotzky and A. Mann [7] had appeared, the importance of this family in the study of finite p-groups became clear. It was observed that the structure of powerful p-groups is also quite similar to that of abelian groups. For example, it was proved in [7] i i that if G is a powerful p-group, then Gp is also powerful and Gp is precisely the set of i p th powers of G. Recently, L. Wilson [8] proved that when p is odd, Ωi(G) (the subgroup generated by the elements of order less than or equal to pi ) is precisely the set of elements i of order less than or equal to p ; even more, he proved that the nilpotency class of Ωi(G) is bounded by i +1 (a short proof of these facts is also given by G. Fernández-Alcober in [3]). This, together with a result of L. Hethelyi and L. Levai [5], gives us that powerful p-groups are power abelian. Another interesting family of groups was introduced by D. Arganbright p in [1]. He proved that if G is a p-group, p is odd and G satisfies that γp−1(G) G ,then it follows that G is p-powered (Gp is precisely the set of p-powers of G). The aim of this paper is to extend these results to other families of finite p-groups. 4 p We say that a finite p-group is potent if [G, G] G for p = 2orγp−1(G) G for p>2. Note that for p = 2andp = 3 to be potent is the same as powerful. In general, any powerful p-group is also potent. The main results of this paper are collected in the next theorem: Theorem 1.1. Let G be a finite potent p-group. (1) If p = 2 then i+1 2i (a) the exponent of Ωi(G) is at most 2 and, even more, [Ωi(G), G] = 2 2i Ωi(G ) = 1; (b) the nilpotency class of Ωi(G) is at most (i + 3)/2; (c) if N G and N G2 then N is power abelian; (d) if N G and N G4 then N is powerful. (2) If p>2 then i (a) the exponent of Ωi(G) is at most p ; (b) the nilpotency class of Ωi(G) is at most (p − 2)i + 1; (c) if N G then N is power abelian; (d) if N G and N Gp then N is powerful. We want to note that a part of the previous theorem is also proved in the PhD thesis of L. Wilson [9] and in [8] using different methods. J. González-Sánchez, A. Jaikin-Zapirain / Journal of Algebra 276 (2004) 193–209 195 2. Preliminaries Before starting with the main part of this paper we will introduce some definitions and recall some properties about p-groups. We will use the notation p = p if p is odd and p/p 2 p = 4ifp = 2 (note that G = G ). We will write [N,kM] for [N,M,...,M],where M appears k times. The symbols r and r will denote the smallest integer greater than or equal to r and the greatest integer less than or equal to r respectively. In the next theorem we collect the basic properties of potent p-groups (see [1,7]). Theorem 2.1. Let G be a potent p-group. Then the following properties hold: 4 p (1) If p = 2,thenγk+1(G) γk(G) , and if p>2 then γp−1+k(G) (γk+1(G)) . (2) γi(G) is potent. (3) x,[G, G] is potent, for all x ∈ G. (4) G is p-powered. (5) If N is a normal subgroup of G then G/N is potent. The next two results are basic facts about finite p-groups. Theorem 2.2. Let G be a finite p-group and N, M normal subgroups of G.IfN M[N,G]Np then N M. Theorem 2.3 (Three Subgroups Lemma). Let G be a p-group, H , J and K subgroups of G and N a normal subgroup of G such that [H,J,K], [K,H,J] N.Then[J,K,H] N; in particular, if N, L are normal subgroups of G,then[N,N,L] [L,N,N]. Corollary 2.4. Let G be a p-group and N a normal subgroup of G.Then[γk(G), N] [N,kG]. The following theorem given by P. Hall will be used several times in the paper. It is known as P. Hall’s collection formula. Theorem 2.5 (P. Hall’s formula). Let G be a group and x,y elements of G.Then pn ≡ pn pn pn pn−1 pn−2 pn−3 ··· (xy) x y mod γ2(L) γp(L) γp2 (L) γp3 (L) γpn (L) where L =x,y, and [ ]pn ≡ pn pn pn−1 pn−2 ··· x,y x ,y mod γ2(M) γp(M) γp2 (M) γpn (M) where M =x,[x,y]. We will need the following corollary which can be proved by induction on k. 196 J. González-Sánchez, A. Jaikin-Zapirain / Journal of Algebra 276 (2004) 193–209 Corollary 2.6. Let G be a group and x1,...,xk elements of G.Then ··· pn ≡ pn ··· pn pn pn−1 pn−2 pn−3 ··· (x1 xk) x1 xk mod γ2(L) γp(L) γp2 (L) γp3 (L) γpn (L) where L =x1,...,xk. In particular, we have pn pn pn−1 pn−2 pn−3 ··· Ωi Ωi−kγ2(Ωi) γp(Ωi) γp2 (Ωi) γp3 (Ωi) γpn (Ωi), where Ωl = Ωl(G) for l 1 and Ωl ={1} for l 0. 3. Potently embedded subgroups Let G be a p-group and N a normal subgroup. We say that N is potently embedded in 4 p G if [N,G] N for p = 2and[N,p−2G] N for p odd. In this section we will study potently embedded subgroups of potent p-groups. Since when p = 2 a potent 2-group is powerful and all results that we will prove in this section are known for powerful p-groups, we will only consider potent p-groups for odd p. The following theorem is a generalization of Shalev’s Interchanging Lemma (see [6]). Theorem 3.1. Let N and M be potently embedded subgroups of G.Then[Np,M]= [N,M]p. Proof. By P. Hall’s formula we have that p p p p N ,M [N,M] [M,pN] [N,M] M ,N ,N . Now we apply again P. Hall’s formula to [Mp,N] and we get that Np,M [N,M]p Np,M,M,N . Therefore, by Theorem 2.2, [Np,M] [N,M]p. In order to prove the converse, we can assume that [Np,M]=[[N,M]p,G]=1. p Note that since M is potently embedded in G, [M,pN] [M ,N,N]. By the previous argument, [Mp,N] [M,N]p,so p [M,pN] [M,N] ,N = 1. p Now, let us prove by the reverse induction on k that [N,M,γk(N)] = 1forallk 1. p This is clear when k is big enough.