compact spaces with group structure

Proposition. Assume that $(G,M)$ is a group (with multiplication $M:G\times G\to G$) and $G$ is also a topological space. If $G$ is compact Hausdorff and $M:G\times G\to G$ is continuous, then $(G,M)$ is a topological group.

Proof. Indeed, all we need to show is that function $f:G\to G$ given by $f(g)=g^{-1}$ is continuous. Note, that the following holds for the graph of $f$:

$$\mathrm{\Gamma }(f)=\{(g,f(g))\in G\times G\}=\{(g,g^{-1})\in G\times G\}=M^{-1}(e),$$

where $e$ denotes the neutral element in $G$. It follows (from continuity of $M$) that $\mathrm{\Gamma }(f)$ is closed in $G\times G$. It is well known (see the parent object for details) that this implies that $f$ is continuous, which completes the proof. $\mathrm{\square }$