A stronger form of Neumanns BFC-theorem

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 $xin K$. Let $H=langle K^Grangle$ 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.