Finite Lazard Lie group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and Lazard Lie group
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Definition

A finite Lazard Lie group is defined as a group $G$ satisfying the following equivalent conditions:

No.	Shorthand	Explanation
1	finite and a Lazard Lie group	$G$ is a finite group and there is a natural number $c$ such that the 3-local nilpotency class of $G$ is at most $c$ and the order of $G$ does not have any prime divisors less than or equal to $c$.
2	product of its Sylow subgroups, each of which is a Lazard Lie group	$G$ is a finite nilpotent group, i.e., it is the internal direct product of its Sylow subgroups, and each of its Sylow subgroups is a Lazard Lie group. Here, a finite $p$-group is a Lazard Lie group if its 3-local nilpotency class is at most $p$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group	any two elements commute	Special case $c=1$		|FULL LIST, MORE INFO
odd-order class two group (also called finite Baer Lie group)	odd-order group of class at most two	Special case $c=2$		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite nilpotent group				|FULL LIST, MORE INFO