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:

$N$ and $H$ are normal,
$N\cap H$ is the trivial subgroup $\{e\}$ ,
$NH = G$ (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).