Some Results About The Linear Representations Of Product Groups And Their Applications

The related property of the linear representations of the product of two finite groups is a important direction of the study of representation theory of groups.Studying product of groups will help us to understand properties of groups.Judging from the current developing trends of group representation theory,it is a frontier subject worthy of discussion to extend the relevant conclusions in finite groups to countable infinite groups.Therefore,it will be a very meaningful work to extend the related results of the linear representation of the product group in the finite group to the countable infinite group.In the literature [1],Serre gives a classic theorem: The irreducible representation of the product group of two finite groups is the tensor product of the irreducible representations of these two finite groups.This dissertation mainly promotes this conclusion: Regarding the irreducible representation of the product group of two countable infinite groups,plus a natural condition,the irreducible representation of the product group of two countable infinite groups will be equal to the irreducible representation of the two countable infinite groups represented tensor product.Specifically,the second chapter of this dissertation starts from the basic concept of group representation,gives the definition of product group and the tensor product of representation,as well as some other related property theorems.On this basis,a useful conclusion in finite groups is summarized.Unfortunately,this conclusion is not necessarily true in countable infinite groups.Therefore,in the third chapter of this dissertation,we added a natural condition in the process of popularizing and proving the theorem to ensure the correctness of the theorem's extension.At the same time,We have also obtained an inference about the irreducible representation of the group and the restriction on its subgroups.