3-local nilpotency class

Definition

Suppose $G$ is a group. The 3-local nilpotency class is defined as the 3-local nilpotency class of $G$. Explicitly, it is the supremum, over 3-generated subgroups $H$ of $G$, of the 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

Particular cases

3-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 group of nilpotency class two.
3	Unclear, but the corresponding notion for Lie rings is clear: for a Lie ring, having 3-local nilpotency class at most three is equivalent to having (global) nilpotency class at most three. See nilpotency class three is 3-local for Lie rings.