The Generalized Tensor Dihedral Group Represented Integral Solution

Group representation theory has developed rapidly in modern mathematics and it is a quite active branch of mathematics. It includes ordinary representation theory of finite group, model theory and the whole theory. Among them, ordinary representation theory of finite group was founded earliest. It has been one hundred years of history and its development is the most perfect. It is the basis of the study other group representation theory.Group representation theory, especially indicators (characters) theory of groups, is one of the most powerful tools to study finite groups. The finite group ordinary indicators (i.e., ordinary characters) are first introduced by G. Frobenius since1896, and then the development of the finite group character theory and complex theory is to the point of fairly perfect because of Fro-benius and W. Burnside. They also give the ordinary characters of finite groups representation theory’s application to the finite groups structure theory. Burnside, for example, about the order of the group of solvability theorem and Frobenius of true, a sufficient condition for the existence of normal subgroup. In1905,I. Schur by later generations called Schur lemma for the tool, made great simplification of complex theory which is established by Frobenius and Burnside. They all used the methods of matrix representation and ordinary character. In the century20s, E. Noether used finite dimensional associative algebra structure theory and the module theory, to unified treat the theory of finite dimensional representations theory and representations theory of finite dimensional semisimple associative algebras, thus make the ordinary representations theory of finite dimensional more concise and beautiful. Mathematical development in this century shows that finite group representation theory, except for the structure of finite groups, was studied in many branches of mathematics, and has important applications on other branches of natural science.In this paper, we study the generalized dihedral group GN,n=<h,t,w|t2=h2N=1,w"=hN, tw=w-1t,ht=th,hw=wh) here N≥1be an odd integer and n≥2, and its tensor product decomposition. This paper is divided into four parts. The first part we mainly introduce two equivalent concepts of representation, tensor product of the vector space, the tensor product of representation and representation ring etc. In the second part, first of all, we introduce that the irreducible representations of the dihedral group and find out the base of representation ring R0(Dn) of the dihedral group. Further more, we have carried on the classification of the n in dihedral group, and give the multiplication of representation ring structure respectively. Finally, the specific example in the case that of n=8is given. In the third part we introduce the concept of generalized dihedral group, and find out the generalized dihedral group’s commutator Sub-group GN,n(1) and its conjugate classes, then classify all irreducible representations of the Genera-lized dihedral group. In the fourth part, we have carried on the discussion of n in generalized dihedral group, and give a multiplication of representation ring structure respectively.