The Finite Groups Of Which The Automorphism Group's Order Is 2～3p～2

The research of finite groups can be divided into group structure and group representation. They all have rich contents. The group structure is about solving all kinds of nonobjective structure questions of finite groups, including the characters of p-groups, the questions whether cyclic groups, abelian groups, nilpotent groups and solvable groups can uniquely decompose into p-groups' direct product, the classification problems of finite simple groups, the extension of finite groups and so on. The building of group representation theory is mainly applied to research finite groups' structures, which contains ordinary representation of finite groups, modular representation of finite groups, representation theory of topology groups and so on.In 1983, Machael began to research the solution of the automorphism groups' equation Aut(X)(?)G, in order to find out what kinds of finite groups can be automorphism groups. His research has aroused the multitudinous group theory expert's interest; this is because researching this question for us to understand the automorphism groups' complete picture has the extreme vital significance. The first step is to solve the problems of the abelian groups' being the automorphism groups of the finite group. Because we can completely solve this step hopefully, its research is extremely attractive.For a given finite group G, the problem of solving the equationAut(X)(?)G, generally speaking, is difficult. Iyer proved that there are finite groups satisfying with the equation at most. The same conclusion applies to the equation |Aut(X)|=n(n odd). Currran got a conclusion: for any odd prime number p, the equation |Aut(X)|=pⁿ(1≤n≤5) has no solution. Flym gave out all the solutions of the equation|Aut(X)|=2⁵. Guiyun Chen researched the groups, whose order are P₁, P₂,..., P_n or pq²(p₁, p₂, ..., p_n, p and q are distinct primes). Shirong Li solved all finite groups X with |Aut(X)|=p²q², 2³p or p³q, where p, q are appropriate primes. Du solved the case when |Aut(X)| equals to 4pq (p≠q odd). In this paper we discuss the equation |Aut(X)|=8p². Through the condition known, in the base of Shirong Li's paper about the equation|Aut(X)|=8p, we analyze from G's two characters nilpotent and non-nilpotent. Then further we get some conclusions about G and rich the achievement in the field of researching finite groups' structure.