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: [email protected] (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
SN - 0021-8693
VL - 280
SP - 537
EP - 576
JO - Journal of Algebra
JF - Journal of Algebra
IS - 2
ER -