Boundedly generated subgroups of finite groups

Abstract

The starting point for this work was the question whether every finite group G contains a two-generated subgroup H such that pi(H) = pi(G), where pi(G) denotes the set of primes dividing the order of G. We answer the question in the affirmative and address the following more general problem. Let G be a finite group and let i(G) be a property of G. What is the minimum number t such that G contains a t-generated subgroup H satisfying the condition that i(H) = i(G)? In particular, we consider the situation where i(G) is the set of composition factors (up to isomorphism), the exponent, the prime graph, or the spectrum of the group G. We give a complete answer in the cases where i(G) is the prime graph or the spectrum (obtaining that t = 3 in the former case and t can be arbitrarily large in the latter case). We also prove that if i(G) is the exponent of G, then t is at most four

Coprime invariable generation and minimal-exponent groups

“…, g xt t , so G is coprimely invariably generated. In particular, as a corollary of Theorems 1.2 and 1.3, we deduce a result already proved in [18] and [8]: a finite group G contains a 3-generated subgroup H with exp(G) = exp(H) and if G is soluble there exists indeed a 2-generated subgroup H of G with exp(G) = exp(H).…”

On the length of a finite group and of its 2-generator subgroups

“…A deep theorem of Thompson says that G is soluble if and only if every 2-generator subgroup of G is soluble [11] (see also Flavell [4]). A number of recent results reflecting the phenomenon that properties of a finite group are determined by its boundedly generated subgroups can be found in [10,9,2].…”

Exponent preserving subgroups of the finite simple groups

“…In the recent years a series of problems have been investigated, related to the existence of a suitable subgroup H ă G preserving some prescribed property of G. For example, Lucchini, Morigi, and Shumyatsky [6] proved that if G is finite then it always contains a 2-generated subgroup H with πpGq " πpHq, and a 3-generated subgroup H with ΓpGq " ΓpHq; Covato [3] extended these results to profinite groups; Burness and Covato [1] showed which finite simple groups G contain a proper subgroup H with ΓpGq " ΓpHq.…”