Local nilpotency class

Definition
Suppose $$G$$ is a group and $$k$$ is a natural number. The $$k$$-local nilpotency class is defined as the supremum, over all subgroups $$H$$ of $$G$$ generated by sets of size at most $$k$$, of the defining ingredient::nilpotency class of $$H$$. In other words, it is defined as:

$$\sup_{S \subseteq G, |S| \le k} \operatorname{class}(\langle S \rangle)$$

If there is a non-nilpotent subgroup of $$G$$ generated by $$k$$ or fewer elements, then the $$k$$-local nilpotency class is $$\infty$$. The $$k$$-local nilpotency class may also be infinite because, even though all the subgroups generated by at most $$k$$ elements are nilpotent, there is no finite upper bound on their nilpotency class.

Particular cases

 * 2-local nilpotency class is significant because many of the formulas and constructions for nilpotent groups involve two variables.
 * 3-local nilpotency class is significant because the variety of groups is itself 3-local, and most correspondences, such as the Lazard correspondence, rely only on the 3-local behavior.

Facts

 * The $$1$$-local nilpotency class of a nontrivial group is always $$1$$. This is because cyclic implies abelian.
 * For $$k_1 \le k_2$$, the $$k_1$$-nilpotency class is less than or equal to the $$k_2$$-nilpotency class.
 * For any nilpotent group and any $$k$$, the $$k$$-local nilpotency class of a group is bounded by the nilpotency class of the group.
 * If the $$k$$-local nilpotency class of a group is a value $$c < k$$, then the whole group is nilpotent of class $$c$$. In other words, nilpotency of class $$c$$ is $$(c+1)$$-local. For instance, abelianness is 2-local, and nilpotency of class two is 3-local.