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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[normal subgroups|0CB0DX5BVV]] of $G$.\nThe group $G$ is the internal direct product of $H$ and $K$ when $H \\cap K = \\{e\\}$ and $HK = G$.\n\n## Remarks\n\nIn this case, $hk = kh$ for all $h \\in H$ and $k \\in K$, because $hkh^{-1}k^{-1}$ lies in $H \\cap K$.\nThen the map $(h, k) \\mapsto hk$ is an [[isomorphism|08MG7H8JW5]] from the [[direct product|00HF4RJ5QN]] $H \\times K$ to $G$.\nAlso, each element of $G$ is a product $hk$ with $h \\in H$ and $k \\in K$ in exactly one way.\nFor example, $\\mathbb{Z}/6\\mathbb{Z}$ is the internal direct product of $\\{0, 3\\}$ and $\\{0, 2, 4\\}$.\n\n## Explanations\n\n### L1 human\n\nSometimes a group already contains two subgroups that behave like the two factors of a direct product.\nThey meet only in the identity, they fill the group together, and they are both normal.\nThen the group is the direct product of these subgroups, up to isomorphism.\n",
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