The ranks of central factor and commutator groups

TL;DR

This paper investigates the relationship between the rank of a group modulo its center and the rank of its derived subgroup, extending classical results to finite and certain infinite groups.

Contribution

It establishes that the rank of the derived subgroup is bounded by the rank of the group modulo its center, generalizing Schur's theorem to a broader class of groups.

Findings

The rank of G' is bounded by the rank of G/Z(G) in finite groups.

A similar rank-bounding result holds for a large class of infinite groups.

Provides new insights into the structure of groups with finite or large rank conditions.

Abstract

The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite group such that $G/Z(G)$ has rank $r$, then the rank of $G'$ is $r$-bounded. We also show that a similar result holds for a large class of infinite groups.