Internal Direct Product Theorem/General Result/Proof 1

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Theorem

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

By definition, $G$ is the internal group direct product of $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ if and only if the mapping:

$\displaystyle C: \prod_{k \mathop = 1}^n H_k \to G: \map C {h_1, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$

is a group isomorphism from the cartesian product $\struct {H_1, \circ {\restriction_{H_1} } } \times \cdots \times \struct {H_n, \circ {\restriction_{H_n} } }$ onto $\struct {G, \circ}$.


Necessary Condition

Let $G$ be the internal group direct product of $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$.

$(1): \quad$ From Subgroup Product is Internal Group Direct Product iff Surjective, $G = H_1 H_2 \cdots H_n$.
$(2): \quad$ From Internal Group Direct Product is Injective: General Result, $\sequence {H_k}_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups.
$(3): \quad$ From Internal Group Direct Product Isomorphism:
$\forall k \in \set {1, 2, \ldots, n}: H_k \lhd G$

$\Box$


Sufficient Condition

Now 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: General Result, $C$ 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 $\sequence {H_k} _{1 \mathop \le k \mathop \le n}$.

$\blacksquare$


Sources

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