= Direct product of groups
{wiki}
= Direct products of groups
{synonym}
= Group direct product
{synonym}
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> is
$$
(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2).
$$
The <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 $C_2\times C_3$,
$$
(1,2)+(1,2)=(0,1),
$$
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>:
$$
(g,e_H)(e_G,h)=(g,h)=(e_G,h)(g,e_H).
$$
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 https://judsonbooks.org/aata-files/aata-html/isomorph-section-direct-products.html[Thomas Judson, Abstract Algebra: Theory and Applications, §9.2, Direct Products].
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