Lazard Lie group: Difference between revisions

Revision as of 19:54, 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]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Lazard Lie group, all facts related to Lazard Lie group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

A group $G$ is termed a Lazard Lie group if there is a natural number $c$ such that both the following hold:

No.	Shorthand for property	Explanation
1	$G$ is powered for the set of all primes less than or equal to $c$ .	$G$ is uniquely $p$ -divisible for all primes $p\leq c$ . In other words, if $p\leq c$ is a prime and $g\in G$ , there is a unique value $h\in G$ satisfying $h^{p}=g$ .
2	The 3-local nilpotency class of $G$ is at most $c$ .	For any three elements of $G$ , the subgroup of $G$ generated by these three elements is a nilpotent group of nilpotency class at most $c$ .

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as $c$ increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase $c$ . Thus, a particular value of $c$ may work for a group but larger and smaller values may not.

A Lazard Lie group is a group that can participate on the group side of the Lazard correspondence. The Lie ring on the other side is its Lazard Lie ring.

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 $p-1$ where $p$ is the prime associated with the group.

Relation with other properties

Stronger properties

Abelian group

Weaker properties

Locally nilpotent group

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

@@ Line 5: / Line 5: @@
 ==Definition==
-A [[group]] <math>G</math> is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that ''both'' the following hold:
+A [[group]] <math>G</math> is termed a '''Lazard Lie group''' if there is a natural number <math>c</math> such that '''both''' the following hold:
-# <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>
+{| class="sortable" border="1"
-# 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>.
+! No. !! Shorthand for property !! Explanation
+|-
+| 1 || <math>G</math> is [[defining ingredient::powered group for a set of primes|powered for the set]] of all primes less than or equal to <math>c</math>. || <math>G</math> is uniquely <math>p</math>-divisible for all primes <math>p \le c</math>. In other words, if <math>p \le c</math> is a prime and <math>g \in G</math>, there is a unique value <math>h \in G</math> satisfying <math>h^p = g</math>.
+|-
+| 2 || The [[defining ingredient::local nilpotency class|3-local nilpotency class]] of <math>G</math> is at most <math>c</math>. || 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>.
+|}
+Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as <math>c</math> increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase <math>c</math>. Thus, a particular value of <math>c</math> may work for a group but larger and smaller values may not.
 A Lazard Lie group is a group that can participate on the ''group'' side of the [[Lazard correspondence]]. The Lie ring on the other side is its [[Lazard Lie ring]].