## 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.

Original language | English |
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Pages (from-to) | 537-576 |

Number of pages | 40 |

Journal | Journal of Algebra |

Volume | 280 |

Issue number | 2 |

DOIs | |

State | Published - 15 Oct 2004 |

### Bibliographical note

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.