فهرست مطالب
International Journal of Group Theory
Volume:9 Issue: 1, Mar 2020
- تاریخ انتشار: 1398/12/27
- تعداد عناوین: 7
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Pages 1-6
For a finite group group, denote by mathcal V(G) the smallest positive integer k with the property that the probability of generating G by k randomly chosen elements is at least 1/e. Let G be a finite soluble group. {Assume} that for every pin pi(G) there exists G_pleq G such that p does not divide |G:G_p| and {mathcal V}(G_p)leq d. Then {mathcal V}(G)leq d+7.
Keywords: Finite soluble groups, generation of finite groups -
Pages 7-24
A subgroup H of a group G is called malonormal in G if H cap H^x =langle 1rangle for every element x notin N_G(H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every selfnormalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.
Keywords: Malnormal Subgroups, Malonormal Subgroups, Frobenius Group, Locally Graded groups, Generalized Radical Groups -
Pages 25-30
Generalizing the concept of quasinormality, a subgroup H of a group G is said to be 4-quasinormal in G if, for all cyclic subgroups K of G, langle H,Krangle=HKHK. An intermediate concept would be 3-quasinormality, but in finite p-groups - our main concern - this is equivalent to quasinormality. Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups, particularly in finite p-groups. However, even in the smallest case, when H is a 4-quasinormal subgroup of order p in a finite p-group G, precisely how H is embedded in G is not immediately obvious. Here we consider one of these questions regarding the commutator subgroup [H,G].
Keywords: Finite group, Sylow subgroup, abnormal subgroup, seminormal subgroup -
Pages 31-42
In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, emph{Ann. Math.}, textbf{183} (2011) 769--814) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems.
Keywords: Finite group, maximal subgroup, probabilistic generation, primitive group -
Pages 43-57The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups, except those having a given property, are either self-centralizing or self-normalizing.Keywords: Self-centralizing subgroup, self-normalizing subgroup
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Pages 59-68We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.Keywords: Conjugacy classes, characters, products of conjugacy classes, solvability