Lazard Lie group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group]] <math>G</math>is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>, and such that for any three elements of <math>G</math>, the subgroup of <math>G</math> generated by these three elements is a [[nilpotent group]] of [[nilpotency class]] at most <math>c</math>. | A [[group]] <math>G</math> is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>, and such that for any three elements of <math>G</math>, the subgroup of <math>G</math> generated by these three elements is a [[nilpotent group]] of [[nilpotency class]] at most <math>c</math>. | ||
===p-group version=== | ===p-group version=== | ||
Revision as of 00:21, 10 February 2011
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
A group is termed a Lazard Lie group if there is a natural number such that is uniquely -divisible for all primes , and such that for any three elements of , the subgroup of generated by these three elements is a nilpotent group of nilpotency class at most .
p-group version
A p-group is termed a Lazard Lie group if every subgroup of it generated by three elements, has nilpotency class at most where is the prime associated with the group.
Relation with other properties
Stronger properties
Facts
Lazard's theorem gives a method to construct a Lazard Lie ring for any Lazard Lie group. This construction and its paraphernalia go under the name of the Lazard correspondence.
Metaproperties
Subgroups
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties
Quotients
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties
Direct products
This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties
3-local
A group occurs as a Lazard Lie group if and only if, for any three elements of the group, the subgroup they generate occurs as a Lazard Lie group. For full proof, refer: Lazard Lie property is 3-local