# 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 $p \le c$. 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 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 $\le c \le$ 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