Let $G$ and $H$ be two groups. The direct product of $G$ and $H$ is the group $G \times H$ with multiplication defined by
\[(g_1, h_1) (g_2, h_2) = (g_1 g_2, h_1 h_2)\]for all $g_1$, $g_2 \in G$ and $h_1$, $h_2 \in H$.
Let $G$ be a group with neutral element $e$ and $H$ be a group with neutral element $i$.
The neutral element of $G \times H$ is $(e, i)$.
For all $(g, h) \in G \times H$,
\[(g, h)^{-1} = (g^{-1}, h^{-1}).\]