Lazard Lie group

Quick definition
A group is termed a Lazard Lie group if its defining ingredient::3-local nilpotency class is finite and less than or equal to the group's defining ingredient::powering threshold.

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

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

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

Set of possible values $$c$$ for which a group is a class $$c$$ Lazard Lie group
A group is a Lazard Lie group if and only if its 3-local nilpotency class is less than or equal to its powering threshold. The set of permissible $$c$$ values for which the group is a class $$c$$ Lazard Lie group is the set of $$c$$ satisfying:

3-local nilpotency class $$\le c \le$$ powering threshold

p-group version
A p-group is termed a Lazard Lie group if its 3-local nilpotency class is at most $$p - 1$$. In other words, every subgroup of it generated by at most three elements has nilpotency class at most $$p - 1$$ where $$p$$ is the prime associated with the group.

Weaker properties

 * Stronger than::Locally nilpotent group