BFC-theorems for higher commutator subgroups

TL;DR

This paper extends classical results on BFC-groups to groups where conjugacy classes of certain commutator values are bounded, establishing finiteness properties of related verbal subgroups and their derived subgroups.

Contribution

It proves that in groups with bounded conjugacy classes of w-values, the derived subgroup of the verbal subgroup w(G) is finite with bounds depending on the conjugacy class size and the multilinear commutator.

Findings

The derived subgroup of w(G) is finite with size bounded by a function of m and n.

The subgroup [w(w(G)),w(G)] is finite with size bounded by a function of m and n.

Generalization of BFC-group properties to multilinear commutator values.

Abstract

A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w=w(x_1,\dots,x_n) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if x^G has size at most m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If x^{w(G)} has size at most m for every w-value x, then [w(w(G)),w(G)] is finite of order bounded by a function of m and n.