Lazard Lie group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group | A [[p-group]] is termed a '''Lazard Lie group''' if every subgroup of it generated by three elements, has nilpotence class at most <math>p - 1</math> where <math>p</math> is the prime associated with the group. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Weaker than::p-group of nilpotence class less than p]] | |||
* [[Weaker than::Abelian p-group]] | |||
==Facts== | ==Facts== | ||
[[Lazard's theorem]] gives a method to construct a ''Lie algebra'' for any Lazard Lie group. This construction and its paraphernalia go under the name of the [[Lazard correspondence]]. | [[Lazard's theorem]] gives a method to construct a ''Lie algebra'' for any Lazard Lie group. This construction and its paraphernalia go under the name of the [[Lazard correspondence]]. | ||
==Metaproperties== | |||
{{S-closed}} | |||
{{Q-closed}} | |||
{{finite-DP-closed}} | |||
===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. {{proofat|[[Lazard Lie property is 3-local]]}} | |||
Revision as of 22:52, 17 May 2009
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
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Definition
A p-group is termed a Lazard Lie group if every subgroup of it generated by three elements, has nilpotence 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 Lie algebra 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