Group Epimorphism Preserves Subgroups

Theorem

Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.

Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.

Then:

$H \le G_1 \implies \phi \sqbrk H \le G_2$

where:

$\phi \sqbrk H$ denotes the image of $H$ under $\phi$

$\le$ denotes subgroup.

That is, group epimorphism preserves subgroups.

Proof

By definition, $\phi$ is a fortiori a group homomorphism.

The result then follows from Group Homomorphism Preserves Subgroups.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.20 \ \text {(a)}$