Finite groups with three conjugacy class sizes of primary and biprimary elements

Abstract

We determine the structure of finite π(m) -separable groups if the set of conjugacy class sizes of primary and biprimary elements is {1, m, mn} , where m and n are two coprime integers.

A constructive approach: From local subgroups to new classes of finite groups

“…In the present article, we will utilize the S-quasinormal subgroups, which permute with all Sylow subgroups of the group. The topic of conjugacy classes has always been a general topic in group theory (see for example the recent articles [2,3,6,10]). With regard to the number of conjugacy classes of local subgroups, the results in the present article apply to finite groups having exactly one conjugacy class of maximal local subgroup and the present article provides possible approach to exploring finite groups with multiple conjugacy classes of maximal local subgroups.…”