Group Direct Product is a Group

From ProofWiki

Jump to: navigation, search

Theorem

Let $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ be groups whose identites are $e_G$ and $e_H$ respectively.

The group direct product $\left({G \times H, \circ}\right) = \left({G, \circ_1}\right) \times \left({H, \circ_2}\right)$ is a group.

As $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ are both semigroups, $\left({G \times H, \circ}\right)$ is also a semigroup.

Taking the group axioms in turn:

Follows from the closure axiom for a semigroup.

Follows from the associativity axiom for a semigroup.

The identity is $\left({e_G, e_H}\right)$ from External Direct Product Identity.

The inverse of $\left({g, h}\right)$ is $\left({g^{-1}, h^{-1}}\right)$ from External Direct Product Inverses.

$\blacksquare$