Semidirect Product

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

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

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

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 $N\times H$.

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 $N\times H$.