profinite group
Definition
- A group $G$ is profinite if it is isomorphic to an inverse limit of finite groups.
Remark. We could replace the suffix “finite” with other things, like prosolvable, etc.
Examples
1 ( profinite completion)
2 (p-adic integers, see P-adic numbers) $$\mathbb{Z}p=\lim\leftarrow\mathbb{Z}/(p^n),$$ which is also a procyclic group per the remark in the definition.
3 (Galois groups) Every Galois group is profinite. In particular, \begin{gather} \text{Gal}(K/F)\overset{\simeq}{\rightarrow}\lim_{\overset{\leftarrow}{\underset{L/F \text{ finite Galois}}{K/L/F}}}\text{Gal}(L/F) \ \sigma\mapsto (\sigma|_L)\end{gather} To see this, regard the adjunction of categories of subextensions of $K/F$ (take the opposite category, actually) and subgroups of $\text{Gal}(K/F)$, with right adjoint $L/F\mapsto \text{Gal}(K/L)$ and left adjoint $H\mapsto K^H$ ( Fundamental thm of Galois Theory). In direct limit, we have $K/F\cong\underset{\rightarrow}{\lim}L/F$ where we range over $L/F$ finite Galois. By properties of right adjoint in the opposite category, this means that $R(K/F)=1=\underset{\leftarrow}{\lim}\text{Gal}(K/L)$. For each $L/F$ there is an exact sequence $\text{Gal}(K/L)\to G\to \text{Gal}(L/F)$. By property (1) of inverse limit, this means there is an exact sequence $$\underset{\leftarrow}{\lim}\text{Gal}(K/L)=1\to \underset{\leftarrow}{\lim}G=G\to \underset{\leftarrow}{\lim}\text{Gal}(L/F),$$ whence it follows that $$G/1=G=\underset{\leftarrow}{\lim}\text{Gal}(L/F).$$
Properties
Let $G=\underset{\leftarrow}{\lim}G_i$ be a profinite group.
- Consider each $G_i$ with the discrete topology. Then $G$ is
- compact, assuming Tychanoff’s theorem.
- Hausdorff
- totally disconnected
- $G$ is a profinite group iff $G$ is a compact, Hausdorff, totally disconnected topological group.