Inclusion Mapping on Subgroup is Homomorphism

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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 $x \circ_{\restriction H} y \in H$

$\blacksquare$


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