2-local nilpotency class

Definition

Suppose $G$ is a group. The 2-local nilpotency class is defined as the 2-local nilpotency class of $G$. Explicitly, it is the supremum, over 2-generated subgroups $H$ of $G$, of the nilpotency class of $H$. In other words, it is defined as:

$\sup _{x,y\in G}\operatorname {class} (\langle x,y\rangle )$

(Note that $x,y$ 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 two elements, then $G$ is not 2-locally nilpotent. It is also possible that $G$ be non-nilpotent because, while each 2-generated subgroup is nilpotent, there is no upper bound on the nilpotency class. An example is the generalized dihedral group for 2-quasicyclic group.

In general, the 2-local nilpotency class of a nilpotent group is less than or equal to its nilpotency class.

Particular cases

2-local nilpotency class	What it tells us about the group
1	The group is an abelian group. This follows from the observation that abelianness is 2-local.
2	This is the same as being a Levi group (also called a 2-Engel group). For a Lazard Lie group, this is equivalent to having class exactly two. See 2-Engel and Lazard Lie group implies class two.