The $p$-completion of a knot group

Question

In Serre's LNM 5, in the section on profinite groups, he give the exercise:

Let $k$ be a knot in $ \mathbb{R}^3$, and let $G= \pi _1( \mathbb{R}^3-k)$ be the knot group of $k$. Show that the $p$-completion of $G$ is isomorphic to $\mathbb{Z}_p$.

Here $ \pi_1 $ is the fundamental group, and by $p$-adic completion I mean the inverse limit of quotients of $G$ by finite $p$-index normal subgroups.

I'm sort of unsure where to start with this, since I think the problem of actually calculating knot groups can be quite a hard one...

Answer 1

The essential property of knot groups is the following.

The group $G$ has an element $x$ so that every element or $G$ can be written as a product of conjugates of $x$ and $x^{-1}$ (that is, it is normally generated by a single element).

This can be proved by analysis of the Wirtinger presentation.

The essential lemma is the following.

A finite $p$-group which is normally generated by a single element is cyclic.

Proof: Any element $x$ of order a power of p in a finite group is contained in a p-Sylow subgroup, which is a proper normal subgroup, and hence the smallest normal subgroup of $G$ is still a proper subgroup.

It follows that a finite p-group quotient of a knot group is isomorphic to $\Bbb Z/p^k$. Because this is Abelian it is a quotient of the abelianization $H_1(\Bbb R^3 - k) \cong \Bbb Z$, and so it is the p-completion of $\Bbb Z$. Conclude.