Local nilpotency class

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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:

${sup}_{S\subseteq G,\left|S\right|\le k}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}(⟨S⟩)$

If there is a non-nilpotent subgroup of $G$ generated by $k$ or fewer elements, then the $k$-local nilpotency class is $\mathrm{\infty }$. 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.