On boundedly generated subgroups of profinite groups

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.…”

On the Prime Graph of Simple Groups

“…In the same paper, the authors also investigate similar problems for other group invariants, such as π(G) (the set of prime divisors of |G|), ω(G) (the set of orders of elements of G), exp(G) (the exponent of G), etc. For example, [14,Theorem A] implies that every finite group G has a 2-generated subgroup H such that π(G) = π(H), and appropriate extensions to profinite groups have recently been established by Covato [5].…”