Composite of Group Monomorphisms is Monomorphism

Theorem

Let:

$\struct {G_1, \circ}$

$\struct {G_2, *}$

$\struct {G_3, \oplus}$

be groups.

Let:

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

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

be monomorphisms.

Then the composite of $\phi$ and $\psi$ is also a monomorphism.

Proof

A monomorphism is a homomorphism which is also an injection.

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

From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.

$\blacksquare$

Sources

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