On the type of conjugacy classes and the set of indices of maximal subgroups

Document Type: Research Paper

Authors

1 School of Mathematics and Information Science‎, ‎Yantai University‎, ‎Yantai 264005‎, ‎China.

2 Xingjian College of Science and Liberal Arts‎, ‎Guangxi University‎, ‎Nanning 530004‎, ‎China.

Abstract

‎Let $G$ be a finite group‎. ‎By $MT(G)=(m_1,\cdots,m_k)$ we denote the type of‎ ‎conjugacy classes of maximal subgroups of $G$‎, ‎which implies that $G$ has exactly $k$ conjugacy classes of‎ ‎maximal subgroups and $m_1,\ldots,m_k$ are the numbers of conjugates‎ ‎of maximal subgroups of $G$‎, ‎where $m_1\leq\cdots\leq m_k$‎. ‎In this paper‎, ‎we‎ ‎give some new characterizations of finite groups by the type of‎ ‎conjugacy classes of maximal subgroups‎. ‎By $\pi_t(G)$ we denote‎ ‎the set of indices of all maximal subgroups of $G$‎. ‎We also investigate‎ ‎the influence of the set of indices of all maximal subgroups on the structure of‎ ‎finite groups‎.

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