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. A(5)(a), for 1 less than or equal to a less than or equal to 5, a not equal 2. 2. A(8). 3. PSL(3, 4)(e), for 1 less than or equal to e less than or equal to 10. 4. A(5) x PSL(3, 4)(e), for 1 less than or equal to e less than or equal to 10. Based on this result, we virtually show that if G is an ah-group, with pi(G) not subset of or equal to {2, 3, 5, 7}, then F(G) not equal 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 less than or equal to a less than or equal to 5. 2. PSL(3, 4)(e), for 1 less than or equal to e less than or equal to 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) (sic) S-n. These results are of independent interest and are located in appendices for greater autonomy. (C) 2004 Elsevier Inc. All rights reserved.