Bounded Conjugacy Classes, Commutators and Approximate Subgroups

Abstract

Abstract Given a group G, we write gG for the conjugacy class of G containing the element g. A famous theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup Gʹ is finite. We establish the following results: Let $K,n$ be positive integers and G a group having a K-approximate subgroup A. If $|a^G|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle A^G\rangle$ has finite (K, n)-bounded order. If $|[g,a]^G|\leq n$ for all $g\in G$ and $a\in A$, then the commutator subgroup of $[G,A]$ has finite (K, n)-bounded order.