On the groups satisfying the converse of Schur's theorem

Abstract

A famous theorem of Schur states that for a group $G$ finiteness of $G/Z(G)$‎ ‎implies the finiteness of $G'.$ The converse of Schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎Recently‎, ‎Podoski and Szegedy proved the truth of the converse of Schur's theorem for capable groups‎. ‎They also established an explicit bound for the index of the center of such groups‎. ‎This paper is devoted to determine some families of groups among non-capable groups which satisfy the converse of Schur's theorem and at the same time admit the Podoski and Szegedy's bound as the upper bound for the index of their centers‎.