Direct product of groups

A direct product combines two groups into one while keeping their operations in separate coordinates. Its underlying set is the Cartesian product $G\times H$, and its group operation isThe identity element is $(e_G,e_H)$, and the inverse element of $(g,h)$ is $(g^{-1},h^{-1})$. Associativity follows coordinate by coordinate. For finite groups, the multiplication principle gives $|G\times H|=|G||H|$.

For example, write the cyclic group $C_m$ as the additive group of residue classes modulo $m$. In $C_2\times C_3$,since each coordinate reduces with its own integer modulus. A pair can track two cyclic counters that advance independently. Updating the first coordinate does not alter the second, and vice versa.

This independence has an algebraic form. The embedded factor elements always commute:The coordinate maps $(g,h)\mapsto g$ and $(g,h)\mapsto h$ are group homomorphisms, so one can analyze either component separately. This does not make the full product an abelian group when one of its factors is nonabelian: two arbitrary pairs need not commute.

The construction is useful in both directions. It builds a larger group from familiar pieces, while the internal direct product theorem recognizes when an existing group can be separated into such pieces. A cyclic group can sometimes be split this way; whether two finite cyclic factors combine into one cyclic group is decided by the cyclicity of a product of two finite cyclic groups. Definitions and the basic recognition theorems are developed in Thomas Judson, Abstract Algebra: Theory and Applications, §9.2, Direct Products.