On finite groups whose derived subgroup has bounded rank

TL;DR

This paper investigates bounds on the size of the center of finite groups based on the rank of their derived subgroup, providing new inequalities and answering a question posed by Pyber.

Contribution

It establishes new bounds on the center of finite groups with derived subgroup of bounded rank, including for capable groups and p-groups, extending previous results.

Findings

Bound $ ext{Z}(G)$ in terms of $|G'|$ and rank $r$

Capable groups satisfy $ ext{Z}(G) ext{Z}(G')$ bounds

For capable p-groups, the rank of $G/\text{Z}(G)$ is bounded

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 $G'$.