On groups with conjugacy classes of distinct sizes

Abstract

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following: 1. A 5 a , for 1 ⩽ a ⩽ 5 , a ≠ 2 . 2. A 8 . 3. PSL ( 3 , 4 ) e , for 1 ⩽ e ⩽ 10 . 4. A 5 × PSL ( 3 , 4 ) e , for 1 ⩽ e ⩽ 10 . Based on this result, we virtually show that if G is an ah-group with π ( G ) ⊈ { 2 , 3 , 5 , 7 } , then F ( G ) ≠ 1 , or equivalently, that G has an abelian normal subgroup. In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S 3 , then the non-abelian socle of G is either trivial or isomorphic to one of the following: 1. A 5 a , for 3 ⩽ a ⩽ 5 . 2. PSL ( 3 , 4 ) e , for 1 ⩽ e ⩽ 10 . Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z 2 ≀ S n . These results are of independent interest and are located in appendices for greater autonomy.