Global Lazard Lie group

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Definition

Quick definition

A group is termed a global Lazard Lie group if its nilpotency class is finite and less than or equal to the group's powering threshold.

Explicit definition

A group $G$ is termed a global class $c$ Lazard Lie group for some natural number $c$ if both the following hold:

No.	Shorthand for property	Explanation
1	The powering threshold for $G$ is at least $c$, i.e., $G$ is powered for the set of all primes less than or equal to $c$.	$G$ is uniquely $p$-divisible for all primes $p\leq c$. In other words, if $p\leq c$ is a prime and $g\in G$, there is a unique value $h\in G$ satisfying $h^{p}=g$.
2	The nilpotency class of $G$ is at most $c$.	$G$ 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.

A group is termed a global Lazard Lie group if it is a global class $c$ Lazard Lie group for some natural number $c$.

A global Lazard Lie group is a group that can participate on the group side of the global Lazard correspondence. The Lie ring on the other side is its global Lazard Lie ring.

Set of possible values $c$ for which a group is a global class $c$ Lazard Lie group

A group is a global Lazard Lie group if and only if its nilpotency class is less than or equal to its powering threshold. The set of permissible $c$ values for which the group is a global class $c$ Lazard Lie group is the set of $c$ satisfying:

nilpotency class $\leq c\leq$ powering threshold

p-group version

A p-group is termed a global Lazard Lie group if its nilpotency class is at most $p-1$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
rationally powered nilpotent group				|FULL LIST, MORE INFO
p-group of nilpotency class less than p				|FULL LIST, MORE INFO
Baer Lie group	class two global Lazard Lie group			|FULL LIST, MORE INFO
class three Lazard Lie group	class three global Lazard Lie group			|FULL LIST, MORE INFO

Weaker properties

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