Lazard Lie group

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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	$G$ 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 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.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	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

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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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