Given Automorphism Group Order Of A Taxa

Given a finite group, one can determine its automorphism group. However, it does not hold when it comes oppositely. It is also true for the relation between a finite group G and the order of its automorphism group|Aut(G)|. Moreover, given|Aut(G)|, how to find out all finite groups G is not only a interesting but also a difficult topic.In this paper, we mainly discuss the relation between the order of Aut(G) and the structure of G, which is to find out all finite groups G in equation|Aut(G)|=n when integer n is given. We discuss finite group Gin equation|Aut(G)|=8P1P2…Pn·And we find all nilpotent groups with automorphism groups of order8P1P2…Pn(P1<P2<…<Pn odd). Then under the condition that Sylow2-subgroups of G are cyclic and Aut(G) are solvable, we classify finite non-nilpotent groups with automorphism groups of order8p1p2…Pn. Also the finite nilpotent groups with the automorphism groups of order16p1p2…Pn (p1<P2<…<Pn odd) are classified.