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| {{DISPLAYTITLE:Pro-''p'' group}}
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| In [[mathematics]], a '''pro-''p'' group''' (for some [[prime number]] ''p'') is a [[profinite group]] <math>G</math> such that for any [[open set|open]] [[normal subgroup]] <math>N\triangleleft G</math> the [[quotient group]] <math>G/N</math> is a [[p-group|''p''-group]]. Note that, as profinite groups are [[compact space|compact]], the open subgroups are exactly the [[closed set|closed]] subgroups of finite [[index of a subgroup|index]], so that the [[discrete space|discrete]] quotient group is always finite.
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| Alternatively, one can define a pro-''p'' group to be the [[inverse limit]] of an [[inverse system]] of discrete finite ''p''-groups.
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| The best-understood (and historically most important) class of pro-''p'' groups is the [[p-adic number|''p''-adic]] analytic groups: groups with the structure of an analytic [[manifold]] over <math>\mathbb{Q}_p</math> such that group multiplication and inversion are both analytic functions.
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| The work of [[Alexander Lubotzky|Lubotzky]] and Mann, combined with [[Michel Lazard]]'s solution to [[Hilbert's fifth problem]] over the ''p''-adic numbers, shows that a pro-''p'' group is ''p''-adic analytic if and only if it has finite [[Prüfer rank|rank]], i.e. there exists a positive integer <math>r</math> such that any closed subgroup has a topological generating set with no more than <math>r</math> elements.
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| ==Examples==
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| * The canonical example is the [[p-adic number|''p''-adic integers]]
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| :: <math>\mathbb{Z}_{p} = \displaystyle \lim_{\leftarrow} \mathbb{Z}/p^n\mathbb{Z}. </math> | |
| * The group <math>\ GL_{n}( \mathbb{Z}_{p}) </math> of invertible ''n'' by ''n'' [[matrix (mathematics)|matrices]] over <math>\ \mathbb{Z}_{p} </math> has an open subgroup ''U'' consisting of all matrices congruent to the [[identity matrix]] modulo <math>\ p\mathbb{Z}_{p} </math>. This ''U'' is a pro-''p'' group. In fact the ''p''-adic analytic groups mentioned above can all be found as closed subgroups of <math>\ GL_{n}( \mathbb{Z}_{p}) </math> for some integer ''n'',
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| * Any finite [[p-group|''p''-group]] is also a pro-''p''-group (with respect to the constant inverse system).
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| ==See also== | |
| * [[Residual property (mathematics)]]
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| ==References==
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| * {{citation | id={{MathSciNet | id=1152800}} | last1=Dixon | first1=J. D. | last2=du Sautoy | first2=M. P. F. | last3=Mann | first3=A. | last4=Segal | first4=D. | title=Analytic pro-p-groups | publisher=[[Cambridge University Press]] | year=1991 | ISBN=0-521-39580-1 | author2-link=Marcus du Sautoy}}
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| [[Category:Infinite group theory]]
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| [[Category:Topological groups]]
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| [[Category:P-groups]]<!-- Properly, a generalization, but useful to have in the category because closely related. -->
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| {{algebra-stub}}
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