Direct product of groups

In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. 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 ${\displaystyle G\oplus H}$. 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.

Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows:

The resulting algebraic object satisfies the axioms for a group. Specifically:

Then the direct product G × H is isomorphic to the Klein four-group:

Let G and H be groups, let P = G × H, and consider the following two subsets of P:

Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H. If we identify these with G and H, respectively, then we can think of the direct product P as containing the original groups G and H as subgroups.

These subgroups of P have the following three important properties: (Saying again that we identify G′ and H′ with G and H, respectively.)