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 $(g,h)\in G\times H$,
\[(g,h)^{-1} = (g^{-1},h^{-1}).\]