Say that $S$ is a monoid. Define a subset of $S$ by $S^\times := \{a\in S\mid a \text{ has an inverse}\}$
How can we show that $S^\times$ is a group with the same operation? Can we use this to prove that $U_n$ is a group?
$U_n:= \{a\in \mathbb{Z}_n\mid \gcd(a,n)=1\}$