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

Abstract

A subgroup- and quotientgroup closed class D of groups is a Dietzmann class if the normal closure 〈 x G 〉 of an element x of an arbitrary group G is a D -group, provided that 〈x〉∈ D and G induces on 〈 x G 〉 a D -group of automorphisms. For a set π of prime numbers, let F π denote the class of finite, L F π that of locally finite π-groups. For any subgroup- and quotientgroup closed class X with F π⊆ X⊆L F π , let H X denote the class of hyper- X -groups, (H X)C that of groups with H X -conjugacy classes. We show that H X and (H X)C —in particular H F π , (H F π)C and (L F π)C —are Dietzmann classes.