Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left topological semigroup.
- What are the invertible (left/right/both) elements of $\beta(G)$ as a semigroup?
- Is right invertiblility the same as left invertibility? (That is, if $xy=1$ does this mean that $yx=1$?)