Profinite group

In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.

The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d\in \mathbb {N} } such that every group in the system can be generated by $d$ elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems.

To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group.

Important examples of profinite groups are the additive groups of $p$-adic integers and the Galois groups of infinite-degree field extensions.

Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups.

Definition

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

First definition (constructive)

A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In this context, an inverse system consists of a directed set Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (I,\leq ),} an indexed family of finite groups Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{G_{i}:i\in I\},} each having the discrete topology, and a family of homomorphisms Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{f_{i}^{j}:G_{j}\to G_{i}\mid i,j\in I,i\leq j\}} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{i}^{i}} is the identity map on Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{i}} and the collection satisfies the composition property $f_{i}^{j}\circ f_{j}^{k}=f_{i}^{k}$ whenever Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\leq j\leq k.} The inverse limit is the set: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varprojlim G_{i}=\left\{(g_{i})_{i\in I}\in {\textstyle \prod \limits _{i\in I}}G_{i}:f_{i}^{j}(g_{j})=g_{i}{\text{ for all }}i\leq j\right\}} equipped with the relative product topology.

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

Second definition

A profinite group is a Hausdorff, compact and totally disconnected topological group: that is, a topological group that is also a Stone space.

Profinite completion

Given an arbitrary group $G$, there is a related profinite group Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\widehat {G}},} the profinite completion of $G$. It is defined as the inverse limit of the groups Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G/N} , where $N$ runs through the normal subgroups in $G$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \eta :G\to {\widehat {G}}} , and the image of $G$ under this homomorphism is dense in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\widehat {G}}} . The homomorphism $\eta$ is injective if and only if the group $G$ is residually finite (i.e., Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \bigcap N=1} , where the intersection runs through all normal subgroups $N$ of finite index).

The homomorphism $\eta$ is characterized by the following universal property: given any profinite group $H$ and any continuous group homomorphism $f:G\rightarrow H$ where $G$ is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous group homomorphism Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g:{\widehat {G}}\rightarrow H} with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f=g\eta } .

Equivalence

Any group constructed by the first definition satisfies the axioms in the second definition.

Conversely, any group $G$ satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varprojlim G/N} where $N$ ranges through the open normal subgroups of $G$ ordered by (reverse) inclusion. If $G$ is topologically finitely generated then it is in addition equal to its own profinite completion.

Surjective systems

In practice, the inverse system of finite groups is almost always surjective, meaning that all its maps are surjective. Without loss of generality, it suffices to consider only surjective systems since given any inverse system, it is possible to first construct its profinite group Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G,} and then reconstruct it as its own profinite completion.

Examples

• Finite groups are profinite, if given the discrete topology.
• The group of $p$-adic integers $\mathbb {Z} _{p}$ under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ranges over all natural numbers and the natural maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/p^n\Z \to \Z/p^m\Z} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge m.} The topology on this profinite group is the same as the topology arising from the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic valuation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z_p.}
• The group of profinite integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\Z}} is the profinite completion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z.} In detail, it is the inverse limit of the finite groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/n\Z} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1,2,3,\dots} with the modulo maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/n\Z \to \Z/m\Z} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\,|\,n.} This group is the product of all the groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L / K} is a Galois extension, consider the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \operatorname{Gal}(L / K)} consisting of all field automorphisms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} that keep all elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} fixed. This group is the inverse limit of the finite groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(F / K),} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} ranges over all intermediate fields such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F / K} is a finite Galois extension. For the limit process, the restriction homomorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(F_1 / K) \to \operatorname{Gal}(F_2 / K)} are used, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_2 \subseteq F_1.} The topology obtained on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K,} but one cannot (yet) control which field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} will be in this case. In fact, for many fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} one does not know in general precisely which finite groups occur as Galois groups over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K.} This is the inverse Galois problem for a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K.} (For some fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
• The étale 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.
• 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is a closed normal subgroup of a profinite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,} then the factor group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / N} is profinite; the topology arising from the profiniteness agrees with the quotient topology.
• Since every profinite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is compact Hausdorff, there exists a Haar measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,} which allows us to measure the "size" of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,} compute certain probabilities, and integrate functions on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi : G \to H} between profinite groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is continuous as long as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is topologically finitely generated. Indeed, any open subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is of finite index, so its preimage in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is also of finite index, and hence it must be open.
• Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota.} Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota} is bijective and continuous by the above result. Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota^{-1}} is also continuous, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota} is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its algebraic structure.

Ind-finite groups

There is a notion of ind-finite group, which is the conceptual dual to profinite groups; i.e. a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is projective if for every surjective morphism from a profinite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \to G} there is a section Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \to H.}

Projectivity for a profinite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is equivalent to either of the two properties:

• the cohomological dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{cd}(G) \leq 1;}
• for every prime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} the Sylow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -subgroups of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} are free pro-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Procyclic group

A profinite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is procyclic if it is topologically generated by a single element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma;} that is, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \overline{\langle \sigma \rangle},} the closure of the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \sigma \rangle = \left\{\sigma^n: n \in \Z\right\}.}

A topological group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is procyclic if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \cong {\textstyle\prod\limits_{p\in S}} G_p} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ranges over some set of prime numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_p} is isomorphic to either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z_p} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/p^n \Z, n \in \N.}

See also

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

References

• 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 (2007), "On finitely generated profinite groups, I: strong completeness and uniform bounds", Annals of Mathematics, 2nd series, 165 (1): 171–238, arXiv:math.GR/0604399, doi:10.4007/annals.2007.165.171.
• Nikolov, Nikolay; Segal, Dan (2007), "On finitely generated profinite groups, II: products in quasisimple groups", Annals of Mathematics, 2nd series, 165 (1): 239–273, arXiv:math.GR/0604400, doi:10.4007/annals.2007.165.239.
• 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:10.1090/S0002-9939-1974-0325587-3, JSTOR 2039560, Zbl 0281.20031.