Finite Groups With Automorphism Gooup Of Order 2pqr~2

The solutions of automorphism group equations Aut(X)≌G, i.e. the finite groups which serve as automorphism groups of finite groups, incite many group theory experts's interest. On this issue, we need to solve the problem that commutative group act as automorphism group of finite group firstly, it is probability we can resolve this question completely. Secondly, for a given positive integer n, we need to research the structure of finite group G which satisfies automorphism group equation |Aut(G)|=n. In general, solve this problem thoroughly is interesting, however, it is difficult. For some conditions about a positive integer n, there have been some literatures were studied, and obtained many meaningful results.In 1979, Iyer proved that there are at most a finite number of solutions satisfy automorphism group equation Aut(X)≌G, The same conclusion is also correct for equation |Aut(G)|=n. In 1981, Flanny and MacHale obtained the structure of finite groups with |Aut(G)|=pn (n=1,2,3,4) and pq, proved there is not exist an abelian finite group satisfies the automorphism group equation |Aut(X)|=pn (n=5,6,7), where p is an odd prime number. In 1988, Curran proved there is no solution satisfy |Aut(X)|= pn(1≤n≤5) for arbitrary an odd prime number p. Subsequently, Flym gave all the solutions of the equation |Aut(G)|=25. Guiyun Chen gave the finite groups which automorphism groups order are p1p2…pn or pq2(p1.p2,…,Pn,P and q are different prime numbers respectively). Shirong Li studied the finite groups G completely that |Aut(G)|=p2q2,23p or p3q, where p,q are distinct primes. Xianggui Zhong got all the solutions of the equation |Aut(G)|=2pq2 (p>q>2). Ni Du resolved the situation which |Aut(G)|=4pq, where p,q are different prime numbers respectively.In this paper we discuss the circumstance that |Aut(G)|=2pqr2. We analyze group G from nilpotent and non-nilpotent, some conclusions about G are obtained. For the case G is nilpotent,we obtain the classification of finite group G through utilize a profound result in references[16].Theorem 2.1 Let G be a finite nilpotent group with|Aut(G)|=2pqr2,Then G is one of the following:(a)C2pqr2+1,C2(2pqr2+1),2pqr2+1 is a prime;(b)C(2qr2+1)2,C2(2qr2+1)2,2qr2+1 is a prime；(c)C(2pr2+1)2,C2(2pr2+1)2,2pr2+1 is a prime；(d)C(2pq+1)3,C2(2pq+1)3,2pq+1 is a prime.For the case G is non- nilpotent, we maitinly through classified discuss the order of inner automorphism group of G.In this case,the key of the problem is that G have no non-trivial direct factor. About this situation,we obtain some solutions that satisfy conditions.Theorem 2.2 Finite group G with automorphism group of order 2pqr2 is no-nilpotent and G have no non-trivial direct factor,some main results as follows:G=TPQR,T∈Syl2(G),P∈Sylp(G),Q∈Sylq(G),R∈Sylr(G).In this case we show:|P|=p,|T|=2,|Q|=q,z(G)is r-group.Then G has the following properties:(Ⅰ)|Cen(G)|r≤r,|Cen(G)|2,p,q=1.(Ⅱ)|p|=p,|Q|=q,|T|=2,and Z(G)is a r- group.(a)D2(2qr2+1):2qr2+1 is a prime.(b)G1=<a,b|ap=bq=1,ab=ar,rq≡1(mod p)>(c)G2=<a,b|ap=br=1,ab=at,tr≡1(mod p)>;(d)G3=<a,b|apq=b2=1,ab=ar,r2≡1(mod pq)>(e)G4=<a,b,c|ap=bq=c2=[b,c]=1,ab=ar,ac=a-1,δp(r)=q>(f)G5=<a,b,c|ap=br=c2=[b,c]=1,ab=at,ac=a-1,δp(t)=r>;...