Profinite group

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups.[1]

A non-compact generalization of a profinite group is a locally profinite group.

Definition

Profinite groups can be defined in either of two equivalent ways.

First definition

A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.[2] In this context, an inverse system consists of a directed set ${\displaystyle (I,\leq )}$, a collection of finite groups ${\displaystyle {\mathcal {G}}=\{G_{i}:i\in I\}}$, each having the discrete topology, and a collection of homomorphisms ${\displaystyle \{f_{i}^{j}:G_{j}\to G_{i}\mid i,j\in I,i\leq j\}}$ such that ${\displaystyle f_{i}^{i}}$ is the identity on ${\displaystyle G_{i}}$ and the collection satisfies the composition property ${\displaystyle f_{i}^{j}\circ f_{j}^{k}=f_{i}^{k}}$. The inverse limit is the set:

${\displaystyle \lim _{\leftarrow }{G_{i}}=\left\{(g_{i})_{i\in I}\in \prod _{i\in I}G_{i}:f_{i}^{j}(g_{j})=g_{i}{\text{ for all }}j\geq i\right\}}$

equipped with the relative product topology. In categorical terms, this is a special case of a cofiltered limit construction. One can also define the inverse limit in terms of a universal property.

Second definition

A profinite group is a Hausdorff, compact, and totally disconnected topological group:[3] that is, a topological group that is also a Stone space. Given this definition, it is possible to recover the first definition using the inverse limit ${\displaystyle \lim _{\leftarrow }{G/N}}$ where ${\displaystyle N}$ ranges through the open normal subgroups of ${\displaystyle G}$ ordered by (reverse) inclusion.

Examples

• Finite groups are profinite, if given the discrete topology.
• The group of p-adic integers ${\displaystyle \mathbb {Z} _{p}}$ under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ where n ranges over all natural numbers and the natural maps ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \to \mathbb {Z} /p^{m}\mathbb {Z} }$ for ${\displaystyle n\geq m}$ are used for the limit process. The topology on this profinite group is the same as the topology arising from the p-adic valuation on ${\displaystyle \mathbb {Z} _{p}}$.
• The group of profinite integers ${\displaystyle {\widehat {\mathbb {Z} }}}$ is the inverse limit of the finite groups ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ where ${\displaystyle n=1,2,3,\dots }$ and we use the maps ${\displaystyle \mathbb {Z} /n\mathbb {Z} \to \mathbb {Z} /m\mathbb {Z} }$ for ${\displaystyle m|n}$ are used in the limit process. This group is the product of all the groups ${\displaystyle \mathbb {Z} _{p}}$, and it is the absolute Galois group of any finite field.
• The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if L/K is a Galois extension, we consider the group G = Gal(L/K) consisting of all field automorphisms of L which keep all elements of K fixed. This group is the inverse limit of the finite groups Gal(F/K), where F ranges over all intermediate fields such that F/K is a finite Galois extension. For the limit process, we use the restriction homomorphisms Gal(F₁/K) → Gal(F₂/K), where F₂ ⊆ F₁. The topology we obtain on Gal(L/K) is known as the Krull topology after Wolfgang Krull. Waterhouse (1974) showed that every profinite group is isomorphic to one arising from the Galois theory of some field K, but one cannot (yet) control which field K will be in this case. In fact, for many fields K one does not know in general precisely which finite groups occur as Galois groups over K. This is the inverse Galois problem for a field K. (For some fields K the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.) Not every profinite group occurs as an absolute Galois group of a field.[4]
• The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in general not profinite: for any prescribed group, there is a 2-dimensional CW complex whose fundamental group equals it (fix a presentation of the group; the CW complex has one 0-cell, a loop for every generator, and a 2-cell for every relation, whose attaching map corresponds to the relation in the "obvious" way: e.g. for the relation abc=1, the attaching map traces a generator of the fundamental groups of the loops for a, b, and c in order. The computation follows by van Kampen's theorem.)
• The automorphism group of a locally finite rooted tree is profinite.

Properties and facts

• Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups. Further, being profinite is an extension property.
• Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If N is a closed normal subgroup of a profinite group G, then the factor group G/N is profinite; the topology arising from the profiniteness agrees with the quotient topology.
• Since every profinite group G is compact Hausdorff, we have a Haar measure on G, which allows us to measure the "size" of subsets of G, compute certain probabilities, and integrate functions on G.
• A subgroup of a profinite group is open if and only if it is closed and has finite index.
• According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group (that is, a profinite group that has a dense finitely-generated subgroup) the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely-generated pro-p groups. The proof uses the classification of finite simple groups.
• As an easy corollary of the Nikolov–Segal result above, any surjective discrete group homomorphism φ: G → H between profinite groups G and H is continuous as long as G is topologically finitely-generated. Indeed, any open subgroup of H is of finite index, so its preimage in G is also of finite index, hence it must be open.
• Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι. Then ι is bijective and continuous by the above result. Furthermore, ι⁻¹ is also continuous, so ι is a homeomorphism. Therefore the topology on a topologically finitely-generated profinite group is uniquely determined by its algebraic structure.

Profinite completion

Given an arbitrary group ${\displaystyle G}$, there is a related profinite group ${\displaystyle {\widehat {G}}}$, the profinite completion of ${\displaystyle G}$.[3] It is defined as the inverse limit of the groups ${\displaystyle G/N}$, where ${\displaystyle N}$ runs through the normal subgroups in ${\displaystyle G}$ of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients). There is a natural homomorphism ${\displaystyle \eta :G\rightarrow {\widehat {G}}}$, and the image of ${\displaystyle G}$ under this homomorphism is dense in ${\displaystyle {\widehat {G}}}$. The homomorphism ${\displaystyle \eta }$ is injective if and only if the group ${\displaystyle G}$ is residually finite (i.e., ${\displaystyle \bigcap {N}=1}$, where the intersection runs through all normal subgroups of finite index). The homomorphism ${\displaystyle \eta }$ is characterized by the following universal property: given any profinite group ${\displaystyle H}$ and any group homomorphism ${\displaystyle f:G\rightarrow H}$, there exists a unique continuous group homomorphism ${\displaystyle g:{\widehat {G}}\rightarrow H}$ with ${\displaystyle f=g\eta }$.

Ind-finite groups

There is a notion of ind-finite group, which is the conceptual dual to profinite groups; i.e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is an ind-group.) The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian torsion groups.

Projective profinite groups

A profinite group is projective if it has the lifting property for every extension. This is equivalent to saying that G is projective if for every surjective morphism from a profinite H → G there is a section G → H.[5][6]

Projectivity for a profinite group G is equivalent to either of the two properties:[5]

• the cohomological dimension cd(G) ≤ 1;
• for every prime p the Sylow p-subgroups of G are free pro-p-groups.

Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries.[7]

Procyclic group

A profinite group ${\displaystyle G}$ is procyclic if it is topologically generated by a single element ${\displaystyle \sigma }$ i.e., ${\displaystyle G={\overline {\langle \sigma \rangle }}}$ of the subgroup ${\displaystyle \langle \sigma \rangle =\left\{\sigma ^{n}:n\in \mathbb {Z} \right\}}$.[8]

A topological group ${\displaystyle G}$ if procyclic iff ${\displaystyle G\cong \prod _{p}G_{p}}$ where ${\displaystyle p}$ ranges over all rational primes and ${\displaystyle G_{p}}$ is isomorphic to either ${\displaystyle \mathbb {Z} _{p}}$ or ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} ,n\in \mathbb {N} }$.[9]

See also

• Locally cyclic group
• Pro-p group
• Profinite integer
• Residual property (mathematics)
• Residually finite group
• Hausdorff completion

References

1. 1944-, Wilson, John S. (John Stuart) (1998). Profinite groups. Oxford: Clarendon Press. ISBN 9780198500827. OCLC 40658188.CS1 maint: numeric names: authors list (link)
2. Lenstra, Hendrik. "Profinite Groups" (PDF). Leiden University.
3. Osserman, Brian. "Inverse limits and profinite groups" (PDF). University of California, Davis. Archived from the original (PDF) on 2018-12-26.
4. Fried & Jarden (2008) p. 497
5. Serre (1997) p. 58
6. Fried & Jarden (2008) p. 207
7. Fried & Jarden (2008) pp. 208,545
8. Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-03983-0. ISBN 978-3-642-08473-7.
9. "MO. decomposition of procyclic groups". MathOverflow.

• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Nikolov, Nikolay; Segal, Dan (2006). "On finitely generated profinite groups. I. strong completeness and uniform bounds". arXiv:math.GR/0604399..
• Nikolov, Nikolay; Segal, Dan (2006). "On finitely generated profinite groups. II. products in quasisimple groups". arXiv:math.GR/0604400..
• Lenstra, Hendrik (2003), Profinite Groups (PDF), talk given at Oberwolfach.
• Lubotzky, Alexander (2001), "Book Review", Bulletin of the American Mathematical Society, 38 (4): 475–479, doi:10.1090/S0273-0979-01-00914-4. Review of several books about profinite groups.
• Serre, Jean-Pierre (1994), Cohomologie galoisienne, Lecture Notes in Mathematics (in French), 5 (5 ed.), Springer-Verlag, ISBN 978-3-540-58002-7, MR 1324577, Zbl 0812.12002. Serre, Jean-Pierre (1997), Galois cohomology, Translated by Patrick Ion, Springer-Verlag, ISBN 3-540-61990-9, Zbl 0902.12004
• Waterhouse, William C. (1974), "Profinite groups are Galois groups", Proceedings of the American Mathematical Society, American Mathematical Society, 42 (2): 639–640, doi:10.2307/2039560, JSTOR 2039560, Zbl 0281.20031.