 Closed subsemigroup of a compact group

A nonempty closed subsemigroup $S$ of a compact group is a subgroup. For $x\in S$ and a neighbourhood $U$ of the identity, choose $V$ with $V^{-1}V\subseteq U$ and cover the compact group by finitely many translates of $V$. Two of sufficiently many powers $1,x,x^2,\dots$ lie in the same translate, so some positive power of $x$ belongs to $U$. Hence the identity is a limit of positive powers and lies in $S$. The preceding powers converge to $x^{-1}$, which also lies in $S$. Applied to the closure of a positive-word monoid, this proves that it equals the closed subgroup generated by its generators, without assuming they already include their inverses.