Internal Semidirect Product - Imported Knowlpedia Content

Let $G$ be a group and let $N,H\le G$ be subgroups. One says that $G$ is the internal semidirect product of $N$ and $H$ if:

$N$ is a normal subgroup,
$N\cap H=\{e\}$ ,
$NH=G$ .

Conjugation by elements of $H$ defines a homomorphism $H\to \operatorname{Aut}(N)$ (coming from the conjugation action), and with respect to this map one has an isomorphism

G \cong N \rtimes H,

so internal semidirect products are precisely the internal realizations of semidirect products.

Examples

$S_3$ is an internal semidirect product of $A_3$ (normal, order $3$ ) and $\langle(12)\rangle$ (order $2$ ), hence $S_3\cong C_3\rtimes C_2$ .
$D_{2n}$ is an internal semidirect product of its rotation subgroup $C_n$ and a reflection subgroup $C_2$ .
If, in addition, $H$ is normal, then $G$ is an internal direct product.