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.


1. What are the invertible (left/right/both) elements of $\beta(G)$ as a semigroup?
2. Is right invertiblility the same as left invertibility? (That is, if $xy=1$ does this mean that $yx=1$?)



Stone–Čech compactification as a semigroup
Corollary 4.33 of Hindman and Strauss's book on Algebra in the Stone Cech Compactification says that if $S$ is an infinite cancellative (discrete) semigroup, then the nonprincipal ultrafilters in $\beta S$ form a two-sided ideal. In particular every left invertible element is invertible and the units of $\beta S$ and $S$ coincide. If $S$ is a group this means there are no left or right invertible elements outside of $S$.


