3-local nilpotency class

Definition
Suppose $$G$$ is a group. The 3-local nilpotency class is defined as the 3-defining ingredient::local nilpotency class of $$G$$. Explicitly, it is the supremum, over 3-generated subgroups $$H$$ of $$G$$, of the defining ingredient::nilpotency class of $$H$$. In other words, it is defined as:

$$\sup_{x,y,z \in G} \operatorname{class}(\langle x,y,z \rangle)$$

(Note that $$x,y,z$$ are allowed to be equal to each other, but this does not matter for nontrivial groups).

If there is a non-nilpotent subgroup of $$G$$ generated by three or fewerelements, then $$G$$ is not 3-locally nilpotent. It is also possible that $$G$$ be non-nilpotent because, while each 3-generated subgroup is nilpotent, there is no upper bound on the nilpotency class. An example is the generalized dihedral group for 2-quasicyclic group.

Note that when we say "a group of 3-local nilpotency class $$c$$" we usually mean "a group whose 3-local nilpotency class is at most $$c$$."

In general, the 3-local nilpotency class of a nilpotent group is less than or equal to its nilpotency class.

Related notions

 * 3-local nilpotency class of a Lie ring