I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first define a profinite group as a Hausdorff, compact topological group such that the open subgroups form a base for the neighbourhoods of the identity.

I would like to give my students a few ‘natural’ examples. I am looking for examples of profinite groups that give the students something they can wrap their heads around without knowing about products of finite groups and inverse limits. Ideally, I am looking for examples which have more of a topological emphasis.

As an example of what I am looking for: The $p$-adic integers $\mathbb{Z}_p$ can be constructed as the completion of $\mathbb{Z}$ with the metric space structure induced from the $p$-adic absolute value.

I would particularly like to see an example of a group acting on some object which induces some topology on the group. I would like to exclude Galois groups: these will be covered in another part of the course. I would also like to exclude the fundamental groups encountered in algebraic geometry: I do not expect my students be familiar with this material.

appreciatethe topological viewpoint. $\endgroup$3more comments