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  "body": "\n## Definition\n\nLet $N$ and $H$ be [[groups|G5S1M9]], and let $\\varphi \\colon H \\to \\operatorname{Aut}(N)$, $h \\mapsto \\varphi_h$, be a [[group homomorphism|0JYTG5K9Y4]].\nThe semidirect product $N \\rtimes_\\varphi H$ is the set $N \\times H$ with the operation $(n_1, h_1)(n_2, h_2) = (n_1 \\varphi_{h_1}(n_2), h_1 h_2)$.\n\n## Remarks\n\nThe identity is $(e, e)$, and the inverse of $(n, h)$ is $(\\varphi_{h^{-1}}(n^{-1}), h^{-1})$.\nThe set $\\{(n, e) : n \\in N\\}$ is a [[normal subgroup|0CB0DX5BVV]] isomorphic to $N$, and $\\{(e, h) : h \\in H\\}$ is a subgroup isomorphic to $H$.\nIf $\\varphi$ is the trivial homomorphism, then $N \\rtimes_\\varphi H$ is the [[direct product|00HF4RJ5QN]] $N \\times H$.\nThe [[dihedral group|0JZ27DMJ5D]] $D_n$ is $\\mathbb{Z}/n\\mathbb{Z} \\rtimes_\\varphi \\mathbb{Z}/2\\mathbb{Z}$, where the nonzero element of $\\mathbb{Z}/2\\mathbb{Z}$ acts by $x \\mapsto -x$.\n\n## Explanations\n\n### L1 human\n\nIn a direct product, the two factors do not interact.\nIn a semidirect product, the second factor $H$ twists the first factor $N$ when you multiply.\nThe homomorphism $\\varphi$ says exactly how each element of $H$ twists $N$.\n",
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