Composite of Group Epimorphisms is Epimorphism

Theorem

Let:

$\struct {G_1, \odot}$

$\struct {G_2, *}$

$\struct {G_3, \oplus}$

be groups.

Let:

$\phi: \struct {G_1, \odot} \to \struct {G_2, *}$

$\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$

be (group) epimorphisms.

Then the composite of $\phi$ and $\psi$ is also a (group) epimorphism.

Proof

A group epimorphism is a group homomorphism which is also a surection.

From Composite of Group Homomorphisms is Homomorphism, $\psi \circ \phi$ is a group homomorphism.

From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surection.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms