Semidirect product

Definition

Let $N$ and $H$ be groups, and let $\varphi \colon H \to \operatorname{Aut}(N)$, $h \mapsto \varphi_h$, be a group homomorphism. The semidirect product $N \rtimes_\varphi H$ is the set $N \times H$ with the operation $(n_1, h_1)(n_2, h_2) = (n_1 \varphi_{h_1}(n_2), h_1 h_2)$.

Remarks

The identity is $(e, e)$, and the inverse of $(n, h)$ is $(\varphi_{h^{-1}}(n^{-1}), h^{-1})$. The set $\{(n, e) : n \in N\}$ is a normal subgroup isomorphic to $N$, and $\{(e, h) : h \in H\}$ is a subgroup isomorphic to $H$. If $\varphi$ is the trivial homomorphism, then $N \rtimes_\varphi H$ is the direct product $N \times H$. The dihedral group $D_n$ 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 $x \mapsto -x$.

Explanations

L1 human

In a direct product, the two factors do not interact. In a semidirect product, the second factor $H$ twists the first factor $N$ when you multiply. The homomorphism $\varphi$ says exactly how each element of $H$ twists $N$.