Local nilpotency class

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Local nilpotency class, all facts related to Local nilpotency class) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 nilpotency class of $H$. In other words, it is defined as:

$\sup _{S\subseteq G,|S|\leq k}\operatorname {class} (\langle S\rangle )$

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 $k_{1}\leq k_{2}$, 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.