Abstract

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of Gbeing nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).