Groups with a covering condition on commutators
Eloisa Detomi, Marta Morigi, Pavel Shumyatsky
TL;DR
We study groups $G$ satisfying the $(k,n)$-covering condition for commutators and show that the derived subgroup $G'$ contains a characteristic subgroup $B$ whose index $[G':B]$ and the order $|B'|$ are bounded by a function of the parameters $k$ and $n$. The approach constructs $B$ as the subgroup generated by a finite set of commutators and carries out a detailed inductive and quotient-based analysis to bound the relevant indices and orders. This extends prior results on probabilistically nilpotent groups of class two and connects covering-type phenomena for multilinear commutator words to structural constraints on $G'$. As a consequence, $G'$ admits a nilpotent-like refinement and tighter control over its commutator structure under the covering condition, enriching the understanding of groups with restricted conjugacy dynamics.
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.