Groups Whose Proper Subgroups of Infinite Rank Have Polycyclic Conjugacy Classes

Abstract

A group G is called a PC-group if the factor group G/C G (〈x〉 G ) is polycyclic for each element x of G. It is proved here that if G is a group of infinite rank whose proper subgroups of infinite rank have the property PC, then G itself is a PC-group, provided that G has an abelian non-trivial homomorphic image. Moreover, under the same assumption, a complete classification of minimal non-PC groups is obtained.