  Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing
the element $x$. 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
derived group $G'$ is finite. We establish the following result.
  Let $n$ be a positive integer and $K$ a subgroup of a group $G$ such that
$|x^G|\leq n$ for each $x\in K$. Let $H=\langle K^G\rangle$ be the normal
closure of $K$. Then the order of the derived group $H'$ is finite and
$n$-bounded.
  Some corollaries of this result are also discussed.
