Lazard Lie group: Difference between revisions

Revision as of 00:03, 3 August 2013

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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

Quick definition

A group is termed a Lazard Lie group if its 3-local nilpotency class is finite and less than or equal to the group's powering threshold.

Explicit 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	The powering threshold for $G$ is at least $c$ , i.e., $G$ 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.

Set of possible values $c$ for which a group is a class $c$ Lazard Lie group

A group is a Lazard Lie group if and only if its 3-local nilpotency class is less than or equal to its powering threshold. The set of permissible $c$ values for which the group is a class $c$ Lazard Lie group is the set of $c$ satisfying:

3-local nilpotency class $\leq c\leq$ powering threshold

p-group version

A p-group is termed a Lazard Lie group if its 3-local nilpotency class is at most $p-1$ . In other words, every subgroup of it generated by at most three elements has nilpotency class at most $p-1$ where $p$ is the prime associated with the group.

Metaproperties

Metaproperty name

Satisfied?

Proof

Statement with symbols

subgroup-closed group property

No

Lazard Lie property is not subgroup-closed

It is possible to have a Lazard Lie group

G

and a subgroup

H

of

G

such that

H

is not a Lazard Lie group in its own right.

quotient-closed group property

No

Lazard Lie property is not quotient-closed

It is possible to have a Lazard Lie group

G

and a normal subgroup

H

of

G

such that the quotient group

G/H

is not a Lazard Lie group in its own right.

finite direct product-closed group property

No

Lazard Lie property is not finite direct product-closed

It is possible to have Lazard Lie groups

G_{1}

and

G_{2}

such that the external direct product

G_{1}\times G_{2}

is not a Lazard Lie group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
abelian group	any two elements commute	Precisely the case $c=1$ (see also Lazard correspondence#Particular cases)	\|FULL LIST, MORE INFO
Baer Lie group	uniquely 2-divisible and class at most two	Precisely the case $c=2$ (see also Lazard correspondence#Particular cases	\|FULL LIST, MORE INFO
p-group of nilpotency class less than p		global nilpotency class puts an upper bound on the 3-local nilpotency class	\|FULL LIST, MORE INFO
rationally powered nilpotent group	nilpotent and uniquely divisible for all primes		\|FULL LIST, MORE INFO

Weaker properties

Locally nilpotent group

@@ Line 33: / Line 33: @@
 A [[p-group]] is termed a '''Lazard Lie group''' if its [[defining ingredient::local nilpotency class|3-local nilpotency class]] is at most <math>p - 1</math>. In other words, every subgroup of it generated by at most three elements has [[nilpotency class of a group|nilpotency class]] at most <math>p - 1</math> where <math>p</math> is the prime associated with the group.
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[Lazard Lie property is not subgroup-closed]] || It is possible to have a Lazard Lie group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> is ''not'' a Lazard Lie group in its own right.
+|-
+| [[dissatisfies metaproperty::quotient-closed group property]] || No || [[Lazard Lie property is not quotient-closed]] || It is possible to have a Lazard Lie group <math>G</math> and a [[normal subgroup]] <math>H</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> is ''not'' a Lazard Lie group in its own right.
+|-
+| [[dissatisfies metaproperty::finite direct product-closed group property]] || No || [[Lazard Lie property is not finite direct product-closed]] || It is possible to have Lazard Lie groups <math>G_1</math> and <math>G_2</math> such that the [[external direct product]] <math>G_1 \times G_2</math> is not a Lazard Lie group.
 ==Relation with other properties==