On finite groups whose derived subgroup has bounded rank

Abstract

Let $G$ be a finite group with derived subgroup of rank $r$. We prove that $\gzz\leq |G'|^{2r}$. Motivated by the results of I. M. Isaacs in \cite{isa} we show that if $G$ is capable then $\gz\leq |G'|^{4r}$. This answers a question of L. Pyber. We prove that if $G$ is a capable $p$-group then the rank of $G/\mathbf{Z}(G)$ is bounded above in terms of the rank of $G^{\prime }$.