Central quotient versus commutator subgroup of groups

TL;DR

This paper explores the relationship between the quotient of a group by its center and its commutator subgroup, extending classical results and classifying certain finite groups based on these properties.

Contribution

It generalizes Neumann's partial converse of Schur's theorem and classifies finite groups where the quotient by the center relates to the commutator subgroup's size.

Findings

Extended the converse of Schur's theorem.

Classified finite groups with specific quotient and commutator subgroup sizes.

Proposed open problems related to group structure.

Abstract

In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if $G$ is a finitely generated group with finite commutator subgroup, then $G/\Z(G)$ is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups $G$ such that $|G/\Z(G)| = |\gamma_2(G)|^d$, where $d$ denotes the number of elements in a minimal generating set for $G/\Z(G)$. Some problems and questions are posed in the sequel.