Groups with a covering condition on commutators

TL;DR

This paper investigates groups satisfying a (k,n)-covering condition on commutators, establishing the existence of a characteristic subgroup within the derived subgroup with bounded index and derived subgroup size, extending prior results.

Contribution

It introduces a new (k,n)-covering condition for groups and proves the existence of a characteristic subgroup with bounded properties within the derived subgroup.

Findings

Existence of a characteristic subgroup B in G' with bounded index and size.

Extension of earlier results on groups with covering conditions.

Relevance to probabilistically nilpotent finite groups of class two.

Abstract

Given a group G and positive integers k,n, we let B_n=B_n(G) denote the set of all elements x in G such that |x^G|\leq n, and we say that G satisfies the (k,n)-covering condition for commutators if there is a subset S in G such that |S|\leq k and all commutators of G are contained in the product SB_n. The importance of groups satisfying this condition was revealed in the recent study of probabilistically nilpotent finite groups of class two. The main result obtained in this paper is the following theorem. Let G be a group satisfying the (k,n)-covering condition for commutators. Then G' contains a characteristic subgroup B such that [G':B] and |B'| are both (k,n)-bounded. This extends several earlier results of similar flavour.