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The semidirect product is a construction in group theory generalizing the direct product. It arises as the structure of a group $G$ with a normal subgroup $N$ having a complement $N$.

Let $N$ and $H$ be groups and $\varphi : H\to \operatorname{Aut}(N)$ a homomorphism. The semidirect product $N\rtimes_\varphi H$ is defined as the set $N\times H$ together with the multiplication rule $$ (n_1,h_1)\cdot(n_2,h_2) = (n_1\cdot\varphi(h_1)(n_2), h_1\cdot h_2). $$

See also: Wikipedia

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