Permutation Group,Non-cyclic Graph And Units Of Integral Group Ring

There are three parts in this article: the permutation group of preserving somerelations of finite group, non-cyclic graph of finite group and units of integral groupring of finite cyclic group.The first chapter discusses the preserving problem of finite group. We learn thatall of bijection which preserve some relation of finite group is its permutation group.Then the structure of the permutation group which preserve some relation of somefinite groups as like an, ambn, b1ab, bab were characterized.The second chapter construct the non-cyclic graph of finite groups. We firstlydescribe some of the basic properties of the non-cyclic graph of finite group. Then wegive some finite groups which is characterized by their non-cyclic graph.The third chapter studies the important problem of units characterization in thestudying of group ring. We firstly give a proof which characterizes the units of theintegral group ring of cyclic group which have order p, where p is some prime, andis diferent the Ferraz’s proof. Then we characterize the kernel of restriction map ofmap:ZCpâ†'Z2[ζ] restricted to U(ZCp). We finally give the units of integral groupring of cyclic group which have order2p.