# Inclusion Mapping on Subgroup is Homomorphism

## 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 homomorphism.

## Proof

Let $x, y \in H$.

Then:

 $\displaystyle \map i x \circ \map i y$ $=$ $\displaystyle x \circ y$ Definition of Inclusion Mapping $\displaystyle$ $=$ $\displaystyle x \circ_{\restriction H} y$ as $x, y \in H$ $\displaystyle$ $=$ $\displaystyle \map i {x \circ_{\restriction H} y}$ as $x \circ_{\restriction H} y \in H$

$\blacksquare$