TY - JOUR

T1 - On groups with conjugacy classes of distinct sizes

AU - Arad, Zvi

AU - Muzychuk, Mikhail

AU - Oliver, Avital

N1 - Funding Information:
* Corresponding author. E-mail address: muzy@netanya.ac.il (M. Muzychuk). 1 This author was partially supported by the Israeli Ministry of Absorption. 2 The contribution of this author is a part of his Master thesis at Bar-Ilan University under the supervision of Z. Arad.

PY - 2004/10/15

Y1 - 2004/10/15

N2 - 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. A5a, 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. A5a, 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.

AB - 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. A5a, 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. A5a, 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.

UR - http://www.scopus.com/inward/record.url?scp=17344369558&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2004.03.029

DO - 10.1016/j.jalgebra.2004.03.029

M3 - Article

AN - SCOPUS:17344369558

VL - 280

SP - 537

EP - 576

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -