Lazard Lie group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

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

1. $G$ is uniquely $p$-divisible for all primes $p\le c$
2. For any three elements of $G$, the subgroup of $G$ generated by these three elements is a nilpotent group of nilpotency class at most $c$.

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

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
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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