文档介绍：24p2006Aut(X)～=Gn,|Aut(G)|=nGn1979IyerGAut(X)～=G,|Aut(G)|=n1981FlanneryMacHale|Aut(G)|=pn(n=1,2,3,4)pqGpn(n= 5,6,7)1988Curranp,|Aut(G)|=pn(n= 1,2,3,4,5)Flym|Aut(G)|= 25p1p2· · ·pnpq2p1p2· · ·pn,p, q|Aut(G)|=p2q2,23pp3qp, q|Aut(G)|= 2pq2(p > q >2)|Aut(G)|= 4pqp, q|Aut(G)|= 24p|Aut(G)|=?=QP, Q∈Sylq(G), P∈Sylp(G),|P|=p,Z(G)2-P?GP～=Z3,Z5Z17, Inn(G)<Aut(G).|Inn(G)|= 2p,G(a)D34,(b)< a, b|a16=b3= 1, ba=b?1>,(c)< a, b|a20=b2= 1, b?1ab=a9>,(d)< a, b|a20= 1, b2=a5, b?1ab=a9>,(e)< a, b|a8=b5= 1, ba=b?1>;|Inn(G)|= 22p|G/CG(P)|= 2,QQ/Z(Q)4Q/Z(Q)～=Z2×Z2,Q′= Φ(Q) =Z(Q)IG～=< a, b, c, d|a2=b2=c3=d2= [a, b] = [a, c] = [b, c] = 1, d?1cd=c?1, d?1ad=a, d?1bd=b >.IIFinite groups with automorphism group oforder24pGraduate student: Zhang Fusheng Supervisor: Zhong XiangguiMajor: Basic Mathematics Direction: Group Theory Grade: 2006AbstractWe pay attention to the solution of the automorphism groups equation Aut(X)～=G,in order to ?nd out ?nite groups that can occur as the automorphism group of a ?nitegroup. These has aroused some group theory expert’s interest. The ?rst is to solve theproblems of the abelian groups being the automorphism groups of the ?nite group, we pletely solve this step hopefully. Then we translate it to a new question: for a givenpositive integern, ?nd out all the ?nite groups satisfying with the equation|Aut(G)|= speaking, this is very di?cult. For some special cases for the given positive integern, a long series of papers can now be found dealing with the 1979, Iyer proved that there are ?nite groups satisfying with the equation Aut(X)～=G. The same conclusion applies to the equation|Aut(G)|=n. In 1981, Flanny and MacHalegot the answer of equation|Aut(G)|=pn(n= 1,2,3,4) orpq, proved that no abel ?nitegroup satisfying the automorphism groups equation|Aut(X)|=pn(n= 5,6,7). In 1988,Curran got a conclusion: for any odd primep, the equation|Aut(X)|=pn(1≤n≤5) has no solution. Then, Flym gave all the solutions of th