Inclusion Mapping on Subgroup is Monomorphism

Theorem

Let $\struct {G, \circ}$ be a group.

Let $\struct {H, \circ {\restriction_H} }$ be a subgroup of $G$.

Let $i: H \to G$ be the inclusion mapping from $H$ to $G$.

Then $i$ is a group monomorphism.

Proof

We have:

• Inclusion Mapping on Subgroup is Homomorphism

• Inclusion Mapping is Injection

The result follows by definition of (group) monomorphism.

$\blacksquare$

Sources

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