Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition

Theorem
Let $\struct {G, \odot}$ be a group

Let $\struct {H, \odot}$ and $\struct {K, \odot}$ be normal subgroups of $\struct {G, \odot}$.

Let $\struct {G, \odot}$ be the internal group direct product of $\struct {H, \odot}$ and $\struct {K, \odot}$.

Then:
 * the restriction of the quotient epimorphism $q_H$ to $K$ is an isomorphism from $K$ onto the quotient group $G / H$

and:
 * the restriction of the quotient epimorphism $q_K$ to $H$ is an isomorphism from $H$ onto the quotient group $G / K$.

Proof
Recall that the quotient epimorphism $q_H: G \to G / H$ is defined as:
 * $\forall x \in G: \map {q_H} x = x \odot H$

From Restriction of Homomorphism is Homomorphism:
 * the restriction of $q_H$ to $K$ is a homomorphism.

From Lagrange's Theorem (Group Theory):
 * $\card {G / H} = \card K$

Then we have:

But by definition of internal group direct product, a representation of $g \in G$ in the form $h \odot k$ where $h \in H$ and $k \in K$ is unique.

That is:

Hence the restriction of $q_H$ to $K$ is an injection.

Hence by Injection to Equivalent Finite Set is Bijection, it follows that:
 * $q_H$ is a bijection.

A bijective homomorphism is an isomorphism by definition.

Similarly, the quotient epimorphism $q_K: G \to G / K$ is defined as:
 * $\forall x \in G: \map {q_K} x = x \odot K = K \odot x$

as $K$ is a normal subgroup of $G$

From Restriction of Homomorphism is Homomorphism:
 * the restriction of $q_K$ to $H$ is a homomorphism.

From Lagrange's Theorem (Group Theory):
 * $\card {G / K} = \card H$

Then we have:

But by definition of internal group direct product, a representation of $g \in G$ in the form $h \odot k$ where $h \in H$ and $k \in K$ is unique.

That is:

Hence the restriction of $q_K$ to $H$ is an injection.

Hence by Injection to Equivalent Finite Set is Bijection, it follows that:
 * $q_K$ is a bijection.

A bijective homomorphism is an isomorphism by definition.

Hence the result.