Boundedly generated subgroups of finite groups

Abstract

Abstract. The starting point for this work was the question whether every finite group G contains a two-generated subgroup H such that , where 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 be a property of G. What is the minimum number t such that G contains a t-generated subgroup H satisfying the condition that ? In particular, we consider the situation where 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 is the prime graph or the spectrum (obtaining that in the former case and t can be arbitrarily large in the latter case). We also prove that if is the exponent of G, then t is at most four.