nLab
profinite group

Definition (categorical form)

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

Commentary

This means that it 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.

Definition (inverse limit form)

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

Commentary

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.

Both ways of looking at profinite groups as useful, as sometimes a concept can be easier to understand in one of the settings.

Examples

• A motivating example is the absolute Galois group of a number field.

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

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

• Any finite group is profinite!

• Any group has a profinite completion.

• A profinite group is an internal group object in the category of profinite spaces, and conversely.

Developments from the concept

• 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).

References

A 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:

• Francis Borceux, George Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.

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