Compact simple group which is not a Lie group

Question

What is an example of compact topological group $ G $ which is abstractly simple ($ G $ has no proper nontrivial normal subgroups) but is not a Lie group?

Answer 1

Let $G$ be a compact simple group. Since the connected component $G_0$ of the group $G$ is normal, either $G_0=1$ or $G_0=G$. If $G_0=1$, then $G$ is a totally disconnected group and hence has any small open normal subgroups. So $G$ is finite.

If $G_0=G$, then $G$ is connected and by the Gleason-Yamabe theorem there exists a normal subgroup $H$ such that the factor $G/H$ is a Lie group. Then $G$ is a Lie group.

Thus a compact simple group is either finite or a connected Lie group.

Answer 2

kabenyuk's answer is correct but we don't have to appeal to the Gleason-Yamabe theorem. By the Peter-Weyl theorem a compact (Hausdorff) group $G$ has the property that its finite-dimensional unitary representations separate points; in particular if $G$ is nontrivial it admits a continuous homomorphism $\rho : G \to U(n)$ with nontrivial image. So if $G$ is simple (or even just topologically simple) it is a closed subgroup of $U(n)$ for some $n$ and so must be a Lie group by the closed subgroup theorem.

(Abstractly, pushing this argument a little further shows that compact (Hausdorff) groups are pro-(compact Lie).)