Is the following statement true or not?
A locally compact Hausdorff space $X$ is a group if and only if its Stone-Cech compactification $\beta X$ is a group.
I assume both occurrences of "is a group" mean "is the underlying space of a topological group." Then Henno Brandsma's first comment gives a counterexample, because $\beta\mathbb Z$ is indeed not the underlying space of a topological group. The reason is that it is not homogeneous: The points in $\mathbb Z$ are isolated and the others are not.