2-signalizers of finite simple groups.
Let be a finite group and the set of numbers of elements with the same order in . In this paper, we prove that a finite group is isomorphic to , where is one of the Mathieu groups, if and only if the following hold: (1) , (2) .
Let be a finite group, and let be the set of conjugacy class sizes of . By Thompson’s conjecture, if is a finite non-abelian simple group, is a finite group with a trivial center, and , then and are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In...