Direct product

Direct product
Symbol${\displaystyle \times}$, ${\displaystyle \oplus}$
Size formula${\displaystyle |G\times H| = |G||H|}$
Algebraic properties
Algebraic structureCommutative monoid
AssociativeYes
CommutativeYes
IdentityTrivial group
Uniquely factorizableYes[note 1]

The direct product of two groups ${\displaystyle (G, \cdot)}$ and ${\displaystyle (H, *)}$ is a group whose elements ${\displaystyle G \times H}$ are given by the ordered pairs ${\displaystyle (g, h)}$, with ${\displaystyle g \in G}$ and ${\displaystyle h \in H}$. Its group operation is defined as

${\displaystyle (g_1,h_1)\bullet(g_2,h_2)=(g_1\cdot g_2,h_1*h_2).}$

When both groups are abelian (commutative), so is the direct product. The order of the direct product of two groups is the product of their orders.

The symmetry group of the prism product of two polytopes is the direct product of their symmetry groups.

Notes

1. For finite groups. See the Krull-Remak-Schmidt theorem for a more granular classification.