Direct product of groups

The direct product $G\times H$ has componentwise multiplication. An element $(g,h)$ of finite component orders has order $\mathrm{lcm}(\mathrm{ord}g,\mathrm{ord}h)$.

The restricted direct sum of a family of groups $G_i$ is the subgroup of their direct product of groups consisting of tuples with only finitely many nonidentity coordinates. Multiplication is coordinatewise. If every factor is a finite group, each subgroup that is a finitely generated group lies in a finite subproduct, so the restricted direct sum is a locally finite group. This gives concrete infinite torsion groups without a uniform bound on element orders.

If $H,K⊴G$, $H\cap K=\{1\}$, and $HK=G$, then multiplication defines an isomorphism $H\times K\to G$. Conversely, the two canonical factors of a direct product satisfy these conditions.