The Influence Of Orders Of Automorphism Groups On The Structure Of Finite Groups

Given a group G, the order of its automorphism group is determined by G. Oppositely, we can't claim the precise structure of G when the order of its auto-morphism group is given, which is an interesting and rather difficult question in the research of finite group.In this paper, we mainly make some research on the order of automorphism group of a group G and its influence on G, that is to say, we want to find out all of the finite groups G when the orders of their automorphism groups is m, where m is a given positive integer. There are two sections in this paper, In the first section, we extend the research in the paper [13] made by Du Ni and Li Shirong, and find out the precise structure of finite groups with automorphism groups of order 4p1P2…pn.In the second section, we investigate the structure of finite groups when the power exponent of 2, which is a prime factor of the order of their automorphism groups, is 3.When a group has a automorphism group of order 8pg,it is difficult to find out the precise structure, so the condition that Sylow 2-subgroup of the group is abelian is given in this paper.Finally the structure of all the finite groups is given out, whose automorphism group has the order of 8pq, among which p, q can't be 3,7.