In what sense are compact groups a natural generalization of finite topological groups

Question

According to Wikipedia, "Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion."

In what sense are compact groups "a natural generalization" of finite groups whith discrete topology? Does it mean that all compact groups can be explained as a (co)limit of some sequence of finite groups? And what property do they carry over?

Answer 1

Does it mean that all compact groups can be explained as a (co)limit of some sequence of finite groups?

No. The inverse limit of finite groups is known as profinite group. All of them are compact, but not all compact groups are profinite, because profinite groups are additionally totally disconnected.

While direct limit is actually trivial, since the direct limit of discrete spaces is discrete.

In what sense are compact groups "a natural generalization" of finite groups whith discrete topology?

In the sense that many properties of finite groups also apply to compact groups.

One example comes from the equivariant topology. If a finite group $G$ acts on a set $X$, then for every orbit, say $O\subseteq X$, there exists an equivariant bijection between that orbit and $G/H$ for some subgroup $H\subseteq G$ (more precisely: the stabilizer of $O$). Well, this is true even when $G$ is not finite. In other words $O$ and $G/H$ are $G$-isomorphic in the category of $G$-sets.

The same holds true if we consider $G$ as a finite discrete group and $X$ as any topological space with $G$ continuously acting on it (i.e. a $G$-space). And in that situation our equivariant bijection even becomes an equivariant homeomorphism. And so $O$ is $G$-isomorphic to $G/H$ in the category of $G$-spaces. But here finitness becomes important: such homeomorphism need not exist for arbitrary topological group (meaning the natural bijection need not have continuous inverse). Generally $O$ and $G/H$ need not be $G$-isomorphic, or even homeomorphic (there are counterexamples). But it turns out that they are whenever $G$ is compact and $X$ Hausdorff. Which is a consequence of the classical fact that any continuous bijection on a compact space with Hausdorff range is a homeomorphism.

Many theorems in the equivariant topology are often automatically true if we simply replace "finite" with "compact".