The commutator subgroup of a pair of groups

Abstract

Schur proved that if $G$ is a group such that $G/Z(G)$ is finite, then $G^{\prime }$ is finite. Several authors studied the converse of Schur’s theorem. In this paper, we extend the converse of Schur’s theorem to a pair of groups. Also, we determine the structure of a pairs of groups under some conditions.