Internal Direct Product

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

1. $N$ and $H$ are normal ,
2. $N\cap H$ is the trivial subgroup $\{e\}$,
3. $NH = G$ (i.e. every $g\in G$ can be written as $g=nh$ with $n\in N$ and $h\in H$).

Under these hypotheses, the multiplication map $N\times H\to G$, $(n,h)\mapsto nh$, is a group isomorphism , so $G\cong N\times H$ as a direct product .

Examples:

• In any direct product $N\times H$, the subgroups $N\times\{e\}$ and $\{e\}\times H$ are normal, intersect trivially, and generate the whole group; thus $N\times H$ is an internal direct product in itself.
• In the cyclic group $C_6=\langle a\rangle$, the subgroups $\langle a^3\rangle$ (order $2$) and $\langle a^2\rangle$ (order $3$) satisfy the conditions, so $C_6\cong C_2\times C_3$.
• Non-example: in $S_3$, the subgroup $A_3$ is normal but a subgroup of order $2$ is not normal, so $S_3$ is not an internal direct product of those two subgroups (it is an internal semidirect product instead).