Internal Direct Product Theorem

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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 $n = 2$.

$\blacksquare$