Central quotient versus commutator subgroup of groups

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.