Relational operator: Difference between revisions
→Object identity vs. Content equality: '=' vs '==' in ocaml/smalltalk were inversed |
en>Derek farn Undid revision 582042068 by 125.60.156.207 (talk) |
||
| Line 1: | Line 1: | ||
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]]. | |||
== Definition == | |||
The Prüfer rank of [[pro-p-group]] <math>G</math> is | |||
::<math>\sup\{d(H)|H\leq G\}</math> | |||
where <math>d(H)</math> is the [[rank of an abelian group|rank]] of the abelian group | |||
:<math>H/\Phi(H)</math>, | |||
where <math>\Phi(H)</math> is the [[Frattini subgroup]] of <math>H</math>. | |||
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>. | |||
== Properties == | |||
Those [[profinite group]]s with finite Prüfer rank are more amenable to analysis. | |||
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 | |||
[[p-adic number|p-adic]] [[manifold]] structure. | |||
== References == | |||
{{unref|date=March 2008}} | |||
{{DEFAULTSORT:Prufer Rank}} | |||
[[Category:Infinite group theory]] | |||
Revision as of 15:46, 17 November 2013
In mathematics, especially in the area of 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 sections. The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer.
Definition
The Prüfer rank of pro-p-group is
where is the rank of the abelian group
- ,
where is the Frattini subgroup of .
As the Frattini subgroup of can be thought of as the group of non-generating elements of , it can be seen that will be equal to the size of any minimal generating set of .
Properties
Those profinite groups with finite Prüfer rank are more amenable to analysis.
Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic - that is groups that can be imbued with a p-adic manifold structure.
References
Food Technologist Anton from Oshawa, usually spends time with pursuits which includes beach tanning, best property developers in singapore developers in singapore and rowing. Loves to travel and ended up enthused after planing a trip to Cidade Velha.