Large Characteristic Subgroups with Restricted Conjugacy Classes

Abstract

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class $$\mathfrak {X}$$ is F-characteristic if any group containing an $$\mathfrak {X}$$-subgroup of finite index has also a characteristic subgroup of finite index that belongs to $$\mathfrak {X}$$. Thus the class $$\mathfrak {A}$$ of abelian groups is F-characteristic. The aim of this paper is to prove that many interesting classes of infinite groups are F-characteristic. Moreover, it is shown that the class of free groups and that of free abelian groups are not F-characteristic.