Definition and Properties of Direct Sum Decomposition of Groups¹

In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.