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

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 $m_1leqcdotsleq m_k$‎. ‎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.