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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[groups|G5S1M9]].\nThe direct product $G \\times H$ is the set of ordered pairs $(g, h)$ with $g \\in G$ and $h \\in H$, with the operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$.\n\n## Remarks\n\nThe identity is $(e_G, e_H)$, and the inverse of $(g, h)$ is $(g^{-1}, h^{-1})$.\nIf $G$ and $H$ are finite, then $|G \\times H| = |G| \\cdot |H|$.\nThe [[order|0WE2NHZJ7B]] of $(g, h)$ is the least common multiple of the orders of $g$ and $h$, when both are finite.\nFor example, $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z}$ is cyclic of order $6$, but $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$ is not cyclic.\n\n## Explanations\n\n### L1 human\n\nThe direct product runs two groups side by side.\nAn element is a pair, and you multiply the first parts in $G$ and the second parts in $H$.\nThe two parts never interact.\n",
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