The Study Of Category Group And Its Relative Properties

This article is mainly about the study of category group and its relative properties through the combination of group and category. And they are all studied by the commutative diagram.First, this paper exhibits the associative law, the identity element, inverse ele-ment and some algebraic properties by commutative diagram, and gives the definition of category group in the non empty set G. Then, through the commutative diagram we give some proofs of some relative properties of the category group. And we further give a description of the definition of sub category group, the isomorphism of category group and their properties by means of commutative diagram, and then the proofs of the relative results. Based on the above, the algebraic operation " · " is extended to more general algebraic operation "?", similarly to the above idea, we define a more general category group in the set category G. We give the concept of sub category group and the isomorphism of category group and their related results. At the same time, we also study the automorphism and the inner automorphism of category group and quotient category group and their relative contents.The contents of this paper are all based on the commutative diagram, the ele-ments of category group and the relationship between them are studied by the com-mutative diagram, namely some algebraic properties are abstracted and generalized by the commutative diagram to achieve the goal of studying the same type objects and the relationship between them.