Semidirect Product - Imported Knowlpedia Content

Let $N$ and $H$ be groups, and let

\varphi: H \to \operatorname{Aut}(N)

be a group homomorphism, where $\operatorname{Aut}(N)$ is the automorphism group. The semidirect product of $N$ by $H$ with respect to $\varphi$ , denoted $N\rtimes_{\varphi} H$ , is the set $N\times H$ with multiplication

(n_1,h_1)(n_2,h_2) := \bigl(n_1\,\varphi(h_1)(n_2),\; h_1h_2\bigr).

Examples

The dihedral group $D_{2n}$ is isomorphic to $C_n\rtimes C_2$ , where the nontrivial element of $C_2$ acts on $C_n$ by inversion.
The group of affine transformations $x\mapsto ax+b$ of a field is a semidirect product of the additive group (translations) by the multiplicative group (scalings).
If $N$ is abelian and $\varphi$ is trivial, then $N\rtimes_{\varphi}H$ is just the usual product $N\times H$ .

Equivalent characterizations

(Equivalently, $\varphi$ encodes a group action of $H$ on $N$ by automorphisms.)

Remarks

The subgroup $N\times\{e\}$ is normal in $N\rtimes_{\varphi}H$ , and $(N\rtimes_{\varphi}H)/(N\times\{e\})\cong H$ . If $\varphi$ is trivial (every $h$ acts as the identity automorphism), then $N\rtimes_{\varphi}H$ reduces to the direct product $N\times H$ .