In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on is associative. Identity The direct product has an identity element, namely , where is the identity element of and is the identity element of . Inverses The inverse of an element of is the pair , where is the inverse of in , and is the inverse of in . Examples Let be the group of real numbers under addition. Then the direct product is the group of all two-component vectors under the operation of vector addition: . Let be the group of positive real numbers under multiplication. Then the direct product is the group of all vectors in the first quadrant under the operation of component-wise multiplication . Let and be cyclic groups with two elements each: Then the direct product is isomorphic to the Klein four-group: Elementary properties Algebraic structure Let and be groups, let , and consider the following two subsets of : and . Both of these are in fact subgroups of , the first being isomorphic to , and the second being isomorphic to . If we identify these with and , respectively, then we can think of the direct product as containing the original groups and as subgroups. These subgroups of have the following three important properties: (Saying again that we identify and with and , respectively.) The intersection is trivial. Every element of can be expressed uniquely as the product of an element of and an element of . Every element of commutes with every element of . Together, these three properties completely determine the algebraic structure of the direct product . That is, if