  Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing
the element $g$. A 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 $G'$ 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.
