Groups with a solvable subgroup of prime-power index
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Groups with a solvable subgroup of prime-power index Raimundo Bastos1
· Csaba Schneider2
Received: 29 June 2020 / Accepted: 5 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper we describe some properties of groups G that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2–3). We prove that if G is a non-solvable group that contains a solvable subgroup of index p α (for some prime p), then the quotient G/Rad(G) of G over the solvable radical is asymptotically small in comparison to p α ! (Theorem 4). Keywords Solvable groups · Hall π -subgroups · Solvable radical · Fermat primes · Mersenne primes Mathematics Subject Classification 20D10 · 20D05 · 20D20 · 20E34 · 20F69
1 Introduction In this paper we explore some properties of groups with solvable subgroups of ppower index for some prime p. In the case of finite groups, our condition is equivalent to requiring that the group contains a solvable Hall p -subgroup. There are several well-known results concerning the solvability of finite groups assuming the existence of certain Hall p -subgroups. The most famous of these is Hall’s Theorem that states that a finite group is solvable if and only if it contains Hall p -subgroups for all p [10]. Furthermore, in a finite solvable group, for a fixed p, the Hall p -subgroups are
Communicated by Adrian Constantin.
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Raimundo Bastos [email protected] Csaba Schneider [email protected] http://www.mat.ufmg.br/csaba
1
Departamento de Matemática, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Brasília, DF 70910-900, Brazil
2
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil
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R. Bastos, C. Schneider
conjugate (see [9] or [18, 9.1.7–9.1.8]). Later Wielandt proved that a finite group is solvable if it possesses three solvable subgroups whose indices are pairwise relatively prime (see for instance [19, Lemma 11.25]). In particular, if a finite group G contains solvable Hall p -subgroups for three different primes, then G is solvable. Arad and Ward extended Wielandt’s result to show that if G contains a Hall 2 -subgroup and a Hall 3 -subgroup, then G is solvable [1,3]. More solvability criteria in terms of some (solvable) subgroups can be found in [2,6,11,12,14,16,20]. Fermat primes are prime numbers of the form 2m + 1 with m ≥ 1. To simplify notation, we set (1) π0 = {2, 7, 13} ∪ { p | p is a Fermat prime}. For m ≥ 2, primes of the form 2m − 1 are said to be Mersenne primes. Our first theorem implies, for certain primes p, the solvability of finite groups G under the condition that G has a solvable Hall p -subgroup. Theorem 1 Let G be a finite group, let p be a prime number and assume that G contains a solvable subgroup of index p α for some α ≥ 1. Then G contains a solvable Hall p -subgroup and the following assertions are valid. (1) If p ∈ / π0 , then G is solvable. (2) If p = 7, 13, then the Hall p -subgroups
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