# LCS-Lazard Lie group

## Contents

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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 $\infty$. Explicitly, for any nonnegative integer $k$, let $\gamma_k^{3-loc}(G)$ denote the $k^{th}$ member of the 3-local lower central series of $G$. Then, $\gamma_k^{3-loc}(G)$ 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