|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''profinite groups''' are [[topological group]]s that are in a certain sense assembled from [[finite group]]s; they share many properties with their finite quotients.
| | I'm Harold and I live in a seaside city in northern Brazil, Sao Paulo. I'm 38 and I'm will soon finish my study at Comparative Politics.<br><br>Take a look at my blog post :: [http://www.rxreviewllc.com/blog/?p=181808 How To Get Free Fifa 15 Coins] |
| | |
| A non-compact generalization of a profinite group is a [[locally profinite group]].
| |
| | |
| == Definition ==
| |
| | |
| Formally, a profinite group is a [[Hausdorff space|Hausdorff]], [[Compact space|compact]], and [[totally disconnected]] [[topological group]]: that is, a topological group that is also a [[Stone space]]. Equivalently, one can define a profinite group to be a topological group that is [[isomorphism|isomorphic]] to the [[inverse limit]] of an [[inverse system]] of [[discrete space|discrete]] [[finite group]]s. In [[category theory|categorical]] terms, this is a special case of a [[filtered category|(co)filtered limit]] construction.
| |
| | |
| == Examples ==
| |
| | |
| * Finite groups are profinite, if given the [[discrete topology]].
| |
| | |
| * The group of [[p-adic number|''p''-adic integers]] '''Z'''<sub>''p''</sup> under addition is profinite (in fact [[procyclic]]). It is the inverse limit of the finite groups '''Z'''/''p''<sup>''n''</sup>'''Z''' where ''n'' ranges over all natural numbers and the natural maps '''Z'''/''p''<sup>n</sup>'''Z''' → '''Z'''/''p''<sup>''m''</sup>'''Z''' (''n'' ≥ ''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 '''Z'''<sub>''p''</sup>.
| |
| | |
| * The [[Galois theory]] of [[field extension]]s 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(''F''<sub>1</sub>/''K'') → Gal(''F''<sub>2</sub>/''K''), where ''F''<sub>2</sub> ⊆ ''F''<sub>1</sub>. The topology we obtain on Gal(''L''/''K'') is known as the '''Krull topology''' after [[Wolfgang Krull]]. {{harvtxt|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 group]]s 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.<ref name=FJ497>Fried & Jarden (2008) p.497</ref>
| |
| | |
| * The [[Étale fundamental group|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 group]]s of [[algebraic topology]], however, are in general not profinite.
| |
| | |
| * The automorphism group of a [[locally finite rooted tree]] is profinite.
| |
| | |
| == Properties and facts ==
| |
| | |
| *Every [[direct product of groups|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 set|closed]] subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the [[subspace (topology)|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 of a subgroup|index]].
| |
| *According to a theorem of [[Nikolay Nikolov (mathematician)|Nikolay Nikolov]] and [[Dan Segal]], in any topologically finitely-generated profinite group (that is, a profinite group that has a [[dense set|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 group]]s. 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<sup>−1</sup> 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 ''G'', there is a related profinite group ''G''<sup>^</sup>, the '''profinite completion''' of ''G''. It is defined as the inverse limit of the groups ''G''/''N'', where ''N'' runs through the [[normal subgroup]]s in ''G'' of finite [[subgroup|index]] (these normal subgroups are [[partial order|partially ordered]] by inclusion, which translates into an inverse system of natural homomorphisms between the quotients). There is a natural homomorphism η : ''G'' → ''G''<sup>^</sup>, and the image of ''G'' under this homomorphism is [[dense set|dense]] in ''G''<sup>^</sup>. The homomorphism η is injective if and only if the group ''G'' is [[residually finite group|residually finite]] (i.e.,
| |
| <math>\cap N = 1</math>, where the intersection runs through all normal subgroups of finite index).
| |
| The homomorphism η is characterized by the following [[universal property]]: given any profinite group ''H'' and any group homomorphism ''f'' : ''G'' → ''H'', there exists a unique [[continuous function (topology)|continuous]] group homomorphism ''g'' : ''G''<sup>^</sup> → ''H'' with ''f'' = ''g''η.
| |
| | |
| ==Ind-finite groups==
| |
| | |
| There is a notion of '''ind-finite group''', which is the concept [[dual (category theory)|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 group|locally finite]] if every [[generating set of a group|finitely-generated]] [[subgroup]] is finite. This is equivalent, in fact, to being 'ind-finite'.
| |
| | |
| By applying [[Pontryagin duality]], one can see that [[abelian group|abelian]] profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian [[torsion group]]s.
| |
| | |
| ==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 (category theory)|section]] ''G'' → ''H''.<ref name=S9758>Serre (1997) p.58</ref><ref name=FJ207>Fried & Jarden (2008) p.207</ref>
| |
| | |
| Projectivity for a profinite group ''G'' is equivalent to either of the two properties:<ref name=S9758/>
| |
| * the [[cohomological dimension]] cd(''G'') ≤ 1;
| |
| * for every prime ''p'' the Sylow ''p''-subgroups of ''G'' are 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]].<ref>Fried & Jarden (2008) pp.208,545</ref>
| |
| | |
| ==See also==
| |
| *[[Locally cyclic group]]
| |
| *[[Pro-p group]]
| |
| *[[Residual property (mathematics)]]
| |
| *[[Residually finite group]]
| |
| *[[Hausdorff completion]]
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| * {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}
| |
| *{{cite arxiv
| |
| | last1 = Nikolov | first1 = Nikolay
| |
| | last2 = Segal | first2 = Dan
| |
| | eprint = math.GR/0604399
| |
| | title = On finitely generated profinite groups. I. strong completeness and uniform bounds
| |
| | year = 2006
| |
| | class = math.GR}}.
| |
| *{{cite arxiv
| |
| | last1 = Nikolov | first1 = Nikolay
| |
| | last2 = Segal | first2 = Dan
| |
| | eprint = math.GR/0604400
| |
| | title = On finitely generated profinite groups. II. products in quasisimple groups
| |
| | year = 2006
| |
| | class = math.GR}}.
| |
| *{{citation
| |
| | last = Lenstra | first = Hendrik | author-link = Hendrik Lenstra
| |
| | publisher = talk given at [[Mathematical Research Institute of Oberwolfach|Oberwolfach]]
| |
| | title = Profinite Groups
| |
| | url = http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf
| |
| | year = 2003}}.
| |
| *{{citation
| |
| | last = Lubotzky | first = Alexander | author-link = Alexander Lubotzky
| |
| | journal = [[Bulletin of the American Mathematical Society]]
| |
| | pages = 475–479
| |
| | title = Book Review
| |
| | volume = 38
| |
| | year = 2001
| |
| | doi = 10.1090/S0273-0979-01-00914-4
| |
| | issue = 4}}. Review of several books about profinite groups.
| |
| *{{citation
| |
| | last = Serre | first = Jean-Pierre | author-link = Jean-Pierre Serre
| |
| | mr = 1324577 | zbl=0812.12002 | language=French
| |
| | isbn = 978-3-540-58002-7
| |
| | publisher = [[Springer-Verlag]]
| |
| | series = Lecture Notes in Mathematics
| |
| | title = Cohomologie galoisienne
| |
| | volume = 5
| |
| | edition = 5
| |
| | year = 1994}}. {{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Galois cohomology | others=Translated by Patrick Ion | publisher=[[Springer-Verlag]] | year=1997| isbn=3-540-61990-9 | zbl=0902.12004 }}
| |
| *{{citation
| |
| | last = Waterhouse | first = William C. | authorlink = William C. Waterhouse
| |
| | doi = 10.2307/2039560
| |
| | issue = 2
| |
| | journal = [[Proceedings of the American Mathematical Society]]
| |
| | pages = 639–640
| |
| | title = Profinite groups are Galois groups
| |
| | volume = 42
| |
| | year = 1974
| |
| | jstor = 2039560 | zbl=0281.20031
| |
| | publisher = American Mathematical Society }}.
| |
| | |
| [[Category:Infinite group theory]]
| |
| [[Category:Topological groups]]
| |