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| In [[mathematics]], especially in the area of [[abstract algebra|algebra]] known as [[group theory]], the '''Prüfer rank''' of a [[pro-p group]] measures the size of a group in terms of the ranks of its [[elementary abelian group|elementary abelian]] [[section (group theory)|section]]s. The rank is well behaved and helps to define analytic pro-p-groups. The term is named after [[Heinz Prüfer]].
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| == Definition ==
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| The Prüfer rank of [[pro-p-group]] <math>G</math> is
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| ::<math>\sup\{d(H)|H\leq G\}</math> | |
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| where <math>d(H)</math> is the [[rank of an abelian group|rank]] of the abelian group
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| :<math>H/\Phi(H)</math>, | |
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| where <math>\Phi(H)</math> is the [[Frattini subgroup]] of <math>H</math>.
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| As the Frattini subgroup of <math>H</math> can be thought of as the group of non-generating elements of <math>H</math>, it can be seen that <math>d(H)</math> will be equal to the ''size of any minimal generating set'' of <math>H</math>.
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| == Properties ==
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| Those [[profinite group]]s with finite Prüfer rank are more amenable to analysis.
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| Specifically in the case of finitely generated [[pro-p group]]s, having finite Prüfer rank is equivalent to having an [[open set|open]] [[normal subgroup]] that is [[powerful p-groups|powerful]]. In turn these are precisely the class of [[pro-p group]]s that are [[p-adic number|p-adic]] analytic - that is groups that can be imbued with a
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| [[p-adic number|p-adic]] [[manifold]] structure.
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| == References ==
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| {{unref|date=March 2008}}
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| {{DEFAULTSORT:Prufer Rank}}
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| [[Category:Infinite group theory]]
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Revision as of 11:19, 27 February 2014
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