Profinite groups

group theory

# Profinite groups

## Definition

### Categorical form

###### Definition

A profinite group is a pro-object in the category of finite groups (thus it might more precisely be called a pro-(finite group)).

Equivalently:

###### Definition

A profinite group is an internal group object in the category of profinite sets.

###### Remark

This means that a profinite group is a cofiltered diagram of finite groups, which is thought of as a “formal limit” but the limit is not actually computed. In most cases, the limit would not actually exist in the category of finite groups, and while it would exist in the category of all groups, it would be “wrong” category-theoretically: maps between profinite groups are not the same as maps between their honest limits in Grp.

However, because of Stone duality, it turns out that maps between profinite groups are the same as maps between their honest limits in the category of topological groups, where the finite groups are given the discrete topology. Thus, the category of profinite groups can alternately be defined as the category of topological groups that are filtered inverse limits of finite groups. Moreover, the topological groups that arise in this way can be characterized as those which are Hausdorff, compact, and totally disconnected, giving a more elementary definition. In other words, their underlying topological spaces are profinite.

### Inverse limit form

###### Definition

A profinite group is an inverse limit of a system of finite groups.

The finite groups are considered as compact discrete topological groups and so the inverse limit, as a closed subspace of the compact space that is the product of all those finite groups has the inverse limit topology, hence is, as is said above, a compact Hausdorff, totally disconnected group.

## Examples

###### Example

A finite group is profinite.

Historically a motivating example was:

###### Example

The absolute Galois group of a number field is profinite.

###### Example

For a prime number $l$ the (additive) group of l-adic integers is profinite in that it is the inverse limit $\mathbb{Z}_l=lim\; \mathbb{Z}/l^n\mathbb{Z}$.

###### Example

In SGA1, Grothendieck defined the algebraic fundamental group of a scheme as a profinite group. (This is linked with his work on pro-representable functors.)

###### Example

Any group has a profinite completion, given by taking the projective limit of the group’s quotients by its finite index normal subgroups.

A finite index subgroup of a profinite group is not necessarily open. Here is a standard way to obtain examples of such. Let $G$ be a finite group, and let $G^{\mathcal{U}}$ be its ultrapower with respect to some ultrafilter $\mathcal{U}$ on $\mathbb{N}$. Since the cardinality and group structure of the finite group $G$ is first-order expressible, the Los ultraproduct theorem tells us $G^{\mathcal{U}} \simeq G.$ There is a canonical quotient map

$\prod_{\mathbb{N}} G \overset{\pi_{\mathcal{U}}}{\twoheadrightarrow} G^{\mathcal{U}} \simeq G$

whose kernel $K \overset{\mathrm{df}}{=} \operatorname{ker}(\pi_{\mathcal{U}})$ has finite index in the profinite group $\prod_{\mathbb{N}} G$ because its surjective image has finite index also. Since every ultrafilter contains the cofinite filter, $K$ meets every basic open of the product topology on $\prod_{\mathbb{N}} G$ and so is dense. Since no proper open subset of $\prod_{\mathbb{N}} G$ is dense, $K$ is not open.

A profinite group is called strongly complete if it is isomorphic to its own profinite completion. Since a profinite group $G$ is isomorphic to the projective limit of its quotients by open finite index normal subgroups, a profinite group is strongly complete if and only if its open finite index normal subgroups are cofinal among its finite index normal subgroups, if and only if all of its finite index subgroups are open (see e.g. this MO answer).

## Properties

• The category of profinite groups has nice ‘exactness’ properties. The projective limit of a system of profinite groups is an exact functor, unlike its behaviour on groups themselves. To extend this behaviour beyond (pro)finite groups sometimes pro-localic groups have been used; see progroup.

• Profinite completions have been extended from groups to homotopy types for the analysis of finitary properties of the homotopy type. Various constructions in algebraic geometry lead naturally to profinite homotopy types.

• Subclasses of profinite groups are extensively studied. For instance, if $p$ is a prime number, a pro-$p$ group is a pro-object in the category of $p$-groups.

• Pro-p analytic groups have been introduced as an analogue of Lie groups, with certain rings of formal power series replacing differentiable functions.

## Analogues of the group algebra construction

If $G$ is a profinite or pro-p group, the best replacement for the group algebra of $G$ in this context will be a pseudocompact algebra. This is the completed group algebra defined as the inverse limit of the ordinary group algebras $k[G/U]$ as $U$ varies through the open normal subgroups of $G$. Here the coefficient ring $k$ will be chosen itself to be a pseudocompact ring. As finite rings are pseudocompact, one of the most appropriate choices will be a $k = \mathbb{Z}_p$, the field of $p$ elements; (see the book by Dixon et al, below).

Introductory lectures include

A survey is given in

• Alexander Lubowsky, Book review, BAMS 38(4), pp. 475-479, (pdf)

A fairly recent textbook is

• L. Ribes and P. Zalesskii, 2000, Profinite groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag, Berlin.

For the connections with, amongst other things, Galois theory from a categorical viewpoint:

For the corresponding ‘analytic theory’ see:

• J. Dixon, M. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, volume 61 of Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press 1999.

A standard starting point for the study of the homological properties of the completed group algebra of a profinite group is

• A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966) 442-470, MR202790, doi