On groups with conjugacy classes of distinct sizes

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. , for 1 a 5, a≠2. 2. A8. 3. PSL(3,4)e, for 1 e 10. 4. A5×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 S3, then the non-abelian socle of G is either trivial or isomorphic to one of the following: 1. , 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 Z2 Sn. These results are of independent interest and are located in appendices for greater autonomy.