A Promotion On The M-group

The study of M-groups has been one of the hotspot issue in the theory of finite groups in the last few years. Using theπ-theory (whereπis the set of prime numbers) established by I. M. Isaacs, we introduce the notion of the Mπ-groups. We prove that Mπ-groups is an extension of M-groups and Mp-groups. The main results of the paper are as follows:In the first two chapters, some basic notions and concepts are collected aboutπ-theories of characters, groups and representations.The relations of Mπ-groups and M-groups are discussed in chapter 3. We obtain aπ-nilpotent group is Mπ-groups if and only if it is an M-groups. We give a sufficient condition when a group which contains a Hallπ-subgroup is an M-group.In chapter 4, we give the Galois conjugate of the Bπ-character, and obtain that the number of Galois conjugacy classes of the Bπ-character of the finiteπ-separable group equals to the number of the rational conjugacy classes.A quantity characteristic of groups is given in chapter 5 and we extend the work of A. E. Parks Partly.