Let $(G,*)$, $(H,\diamond)$ be two groups. The direct product of $G$ and $H$ is the group
\[(G\times H,\otimes),\]where $\otimes$ is a binary operation on $G\times H$ defined by
\[(g_1,h_1)\otimes(g_2,h_2) = (g_1*g_2,h_1\diamond h_2)\]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 $i$.
\[(g,h)^{-1} = (g^{-1},h^{-1}).\]
The neutral element of $G\times H$ is $(e,i)$.
For every $(g,h)\in G\times H$,