Internal Direct Product Theorem
Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $H_1, H_2 \le G$.
Then $G$ is the internal group direct product of $H_1$ and $H_2$ if and only if:
- $(1): \quad G = H_1 \circ H_2$
- $(2): \quad H_1 \cap H_2 = \set e$
- $(3): \quad H_1, H_2 \lhd G$
where $H_1 \lhd G$ denotes that $H_1$ is a normal subgroup of $G$.
General Result
Let $G$ be a group whose identity is $e$.
Let $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of subgroups of $G$.
Then $G$ is the internal group direct product of $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ if and only if:
- $(1): \quad G = H_1 H_2 \cdots H_n$
- $(2): \quad \sequence {H_k}_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups
- $(3): \quad \forall k \in \set {1, 2, \ldots, n}: H_k \lhd G$
where $H_k \lhd G$ denotes that $H_k$ is a normal subgroup of $G$.
Proof 1
Sufficient Condition
Let $G$ be the internal group direct product of $H_1$ and $H_2$.
- $(1): \quad$ From Subgroup Product is Internal Group Direct Product iff Surjective, $G = H_1 \circ H_2$.
- $(2): \quad$ From Internal Group Direct Product is Injective, $H_1$ and $H_2$ are independent subgroups of $G$.
- $(3): \quad$ From Internal Group Direct Product Isomorphism, $H_1 \lhd G$ and $H_2 \lhd G$.
$\Box$
Necessary Condition
Let $C: H_1 \times H_2 \to G$ be the mapping defined as:
- $\forall \tuple {h_1, h_2} \in H_1 \times H_2: \map C {h_1, h_2} = h_1 \circ h_2$
Suppose the three conditions hold.
- $(1): \quad$ From Subgroup Product is Internal Group Direct Product iff Surjective, $C$ is surjective.
- $(2): \quad$ From Internal Group Direct Product is Injective, $C$ is injective.
- $(3): \quad$ From Internal Group Direct Product of Normal Subgroups, $C$ is a group homomorphism.
Putting these together, we see that $C$ is a bijective homomorphism, and therefore an isomorphism.
So by definition, $G$ is the internal group direct product of $H_1$ and $H_2$.
$\blacksquare$
Proof 2
A specific instance of the general result, with $n = 2$.
$\blacksquare$