Understanding Hilbert–Smith conjecture.

Question

By a faithful action of a topological group G on a topological manifold M, we mean a continuous injection $G\to \operatorname{Homeo}(M)$ (where $\operatorname{Homeo}(M)$ has the compact open topology). One equivalent way of expressing HilbertSmith Conj is:

Conjecture. There is no faithful action of $\mathbb Z_p$ on any connected $n$-manifold $M$.

My question is: why this is possible? for example $Z_p$ acts on 2-sphere as rotation along the $z-$axis. Why does this action does not count as faithful?

Answer 1

I think you are mistaking $\Bbb Z_p$ (which is actually the group of $p$-adic integers) for $\Bbb Z/p$ (the integers mod $p$). There is indeed a faithful $\Bbb Z/p$-action on $S^2$ given by rotating $2\pi k/p$ radians about the $z$-axis for $k \in \Bbb Z/p$.

The above example of a faithful $\Bbb Z/p$-action on $S^2$ does not contradict the Hilbert-Smith conjecture. A standard way of stating the Hilbert-Smith conjecture is as follows:

If a locally compact topological group $G$ acts faithfully on some connected $n$-manifold $M$, then $G$ is a Lie group.

We are fine with the case of $\Bbb Z/p$ since it is just a discrete Lie group.