Permutation-like Matrix Groups Having A Generalized Maximal Cycle

Let (?) be a group of invertible complex n×n matrices. We call (?) a permutation-like group if every matriX in (?) is similar to a permutation matrix. Grega Cigler has solved the part of the questions whether a permutation-like group is equivalent to a group of permutation matrices.In this paper,we shall give a common result on a permutation-like group which has at least two elements in its center.The main results are following:Theorem0.1.Let (?)(?) GLn(C) be a finite permutation-like group.and the center|C((?))|>1,then the following are equiualent:1(?) is equiualent to a group of permutation matrices.2Under a suit base Bi,exist a Y∈C((?)),such that for every is equiualent to a permutation group,and if Bi=QijBj with Qij(?)(?),thn Pj are not at most an identity matrix,and where Xi=AiP(i),Ai=diag{Cik1(k),Cik2(k),…,Cikm}with P(i)satisfying to F(P(i)=Pi.Theorem0.2.Let K be a prime number and (?) be a permutation-like matrix group containing a generalized-maximal cycle Ckm,and normalizer of the group<C> in (?).Then,in a suitable basis,N is equiualent to a permutation matrix group if only and if satisfies the following.1.The form of X is in corollary3.6,if l=1,i.e.X=diag{A1,A2,...,Am}, where Ai=DiP0with Di=diag{(λk1)i-1,(λk2)i-1,…,(λkm)i-1)and P0∈(?),l≤i≤k.2.The form of X is in theorem3.13,if l≠1,i.e.X=AP where A=diag{P1,P1,…P1}and F(P)=P0corresponding to permutation π∈Sm being π(i)=if-1where i∈Zk.And the order satisfied to|P1|||π|.