LCS-Lazard Lie group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A LCS-Lazard Lie group is a group $G$ satisfying both the following properties:

1. It is a 3-locally nilpotent group, i.e., any three elements of the group generate a nilpotent subgroup.
2. Its 3-local lower central series powering threshold is $\mathrm{\infty }$. Explicitly, for any nonnegative integer $k$, let $\underset{k}{\overset{3-loc}{\gamma }}(G)$ denote the $k^{th}$ member of the 3-local lower central series of $G$. Then, $\underset{k}{\overset{3-loc}{\gamma }}(G)$ is powered over all the primes $p\le k$.

The definition of LCS-Lazard Lie group is somewhat nicer than the definition of Lazard Lie group in that it does not involve a "nilpotency class-specific definition."

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	LCS-Lazard Lie property is not subgroup-closed	It is possible to have a LCS-Lazard Lie group $G$ and a subgroup $H$ of $G$ such that $H$ is not a Lazard Lie group.
quotient-closed group property	No	LCS-Lazard Lie property is not quotient-closed	It is possible to have a LCS-Lazard Lie group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not a LCS-Lazard Lie group.
finite direct product-closed group property	Yes	LCS-Lazard Lie property is finite direct product-closed	If $G_1$ and $G_2$ are LCS-Lazard Lie groups, then the external direct product $G_1\times G_2$ is a LCS-Lazard Lie group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group
Baer Lie group
LCS-Baer Lie group