# Inner-Lazard Lie group

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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## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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 + 1).

### Explicit definition

A group $G$ is termed an inner-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 strictly less than $c$. $G$ is uniquely $p$-divisible for all primes $p < c$. In other words, if $p \le c$ 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.

### p-group version

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