Group of Units of Submonoid is Subgroup

Theorem

Let $T$ be a monoid.

Let $S$ be a submonoid of $T$.

Let $\map G T$ and $\map G S$ be the groups of units of $T$ and $S$ respectively.

Then $\map G S \subseteq \map G T$ and $\map G S$ is a subgroup of $\map G T$.

Proof

Let $x \in \map G S$.

Then $x \in S$ and there exists $y \in S$ such that $x y = y x = e$.

Since $S \subseteq T$, we have $y \in T$.

So $x \in \map G T$.

So we have $\map G S \subseteq \map G T$.

From Group of Units is Group, $\map G S$ is a group.

So $\map G S$ is a subgroup of $\map G T$.

$\blacksquare$