On finite groups whose derived subgroup has bounded rank

Details On finite groups whose derived subgroup has bounded rank On finite groups whose derived subgroup has bounded rank

2007-06-22

arxiv.org/abs/0706.3246v1

Mathematics

Group Theory

Scientific paper

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 $G'$.

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