Alexandroff and Stone Čech compactification of topological groups.

Question

I wonder what conditions i need on a topological group $G$ to say that its Alexandroff compactification $G^*$ is also a topological group, and $G\leq G^*$.

The same question to $\beta G$.

Thanks a lot.

Answer 1

If $\alpha G$ is the Alexandroff compactification of the locally-compact non-compact topological group $G$ and if $G$ is a sub-group of $\alpha G,$ then let $\{y\}=\alpha G$ \ $G.$ Let $1$ be the identity of $G .$ Since $G$ is not compact we have $G\ne \{1\}.$ So take $x\in G $ with $x\ne 1.$ Then $xy\ne y$ so $xy\in G$ so $xy=x'\in G.$ But $x^{-1}\in G$ so $y=x^{-1}x'\in G,$ a contradiction.