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

the restriction of the quotient epimorphism $q_H$ to $K$ be an isomorphism from $K$ onto the quotient group $G / H$

and:

the restriction of the quotient epimorphism $q_K$ to $H$ be an isomorphism from $H$ onto the quotient group $G / K$

Then $\struct {G, \odot}$ is the internal group direct product of $\struct {H, \odot}$ and $\struct {K, \odot}$,

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.9 \ \text{(b)}$