# Groups with Boundedly Finite Conjugacy Classes of Commutators

## TL;DR

This paper investigates groups where conjugacy classes of commutators are finite and bounded, establishing bounds on the finiteness of derived subgroups and the third term of the lower central series.

## Contribution

It extends classical results by providing bounds on derived groups and central series for groups with bounded conjugacy classes of commutators.

## Key findings

- If conjugacy classes of commutators are bounded, the second derived group is finite and bounded.
- Bounded conjugacy classes of commutators imply the third term of the lower central series is finite and bounded.
- Generalizes Neumann's and Wiegold's results to broader classes of groups.

## Abstract

In 1954 B. H. Neumann discovered that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G' is finite. Later (in 1957) Wiegold found an explicit bound for the order of G'. We study groups in which the conjugacy classes containing commutators are finite with bounded size. We obtain the following results. Let G be a group and n a positive integer. If |x^G|<n for any commutator x in G, then the second derived group G" is finite with n-bounded order. If |x^{G'}|<n for any commutator x in G, then the order of \gamma_3(G') is finite and $n$-bounded. Here \gamma_3(G') is the third term of the lower central series of G'.