# 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.

## Key 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.