= Internal direct product theorem
An internal direct product finds independent coordinates already present inside a <group>. Let $H,K$ be <normal subgroups> of $G$ with
$$
H\cap K=\{e\},\qquad HK=G.
$$
Then the multiplication map
$$
\Phi:H\times K\longrightarrow G,\qquad \Phi(h,k)=hk
$$
is a <group isomorphism>. Thus every element of $G$ has a unique factorization into one element of each subgroup. Conversely, the embedded factors $G\times\{e_H\}$ and $\{e_G\}\times H$ inside an external <direct product of groups> satisfy these conditions.
For a concrete example, take the additive <cyclic group> $C_6=\mathbb Z/6\mathbb Z$ and its <subgroups> $H=\{0,3\}$ and $K=\{0,2,4\}$. They are <normal subgroups> because $C_6$ is an <abelian group>. Their intersection is $\{0\}$, and the sums $h+k$ give every <residue class> exactly once. For instance, $1=3+4\bmod6$. The table makes the unique decomposition visible.
\Image[/internal-direct-product-c6-decomposition.png]
{title=Each element of C6 has one decomposition as an element of the subgroup $\{0,3\}$ plus one of $\{0,2,4\}$}
The proof explains each hypothesis. Normality puts the <group commutator> $[h,k]=h^{-1}k^{-1}hk$ in both $H$ and $K$, so it is the <identity element>. Therefore the factors <commute>, and $\Phi$ preserves the <group operation>. The condition $HK=G$ makes it <surjective>. If $hk=e$, then $h=k^{-1}$ belongs to the intersection, forcing both to be the <identity element>; thus its <group kernel> is trivial and it is <injective>.
Normality cannot simply be omitted. In the <symmetric group> $S_3$, take $H=\langle(123)\rangle$ and $K=\langle(12)\rangle$. Their intersection is trivial and $HK=S_3$, but $K$ is not a <normal subgroup>. With <permutations> composed right to left, $(123)(12)=(13)$ while $(12)(123)=(23)$, so the factors do not <commute>. The setwise decomposition is unique, but its multiplication map is not a <group homomorphism> from $H\times K$. An independent-coordinate interpretation requires the group operations to agree, not merely a bijection of the underlying sets.
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