ON THE GROUPS SATISFYING THE CONVERSE OF SCHURʹʹS THEOREM

Abstract. A famous theorem of Schur states that for a group G finiteness of G=Z(G) implies the finiteness of G0: The converse of Schurʹs theorem is an interesting problem which has been considered by some authors. Recently, Podoski and Szegedy proved the truth of the converse of Schurʹs 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 Schurʹs theorem and at the same time admit the Podoski and Szegedyʹs bound as the upper bound for the index of their centers.