On groups with conjugacy classes of distinct sizes |
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Authors: | Zvi Arad Mikhail Muzychuk Avital Oliver |
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Institution: | aDepartment of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel;bDepartment of Computer Sciences and Mathematics, Netanya Academic College, 42365, Netanya, Israel |
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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. , for 1a5, a≠2. 2. A8. 3. PSL(3,4)e, for 1e10. 4. A5×PSL(3,4)e, for 1e10. 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 3a5. 2. PSL(3,4)e, for 1e10. Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z2Sn. These results are of independent interest and are located in appendices for greater autonomy. |
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