Internal Semidirect Product

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:

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

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

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 .