The Dietzmann property of some classes of groups with locally finite conjugacy classes

A subgroup- and quotientgroup closed class of groups is a Dietzmann class if the normal closure xG of an element x of an arbitrary group G is a -group, provided that and G induces on xG a -group of automorphisms. For a set π of prime numbers, let denote the class of finite, that of locally finite π-groups. For any subgroup- and quotientgroup closed class with , let denote the class of hyper- -groups, that of groups with -conjugacy classes. We show that and —in particular , and —are Dietzmann classes.