Local lower central series

Definition

Suppose $k$ is a positive integer greater than or equal to 2 (note that although the definition makes sense for $k=1$, it is uninteresting because all the groups in the series from the second one onward are trivial). The $k$-local lower central series of a group $G$, is defined as follows: for any positive integer $i$, the $i^{th}$ member of the series, denoted $\gamma _{i}^{k-loc}(G)$, is defined as the subgroup:

$\gamma _{i}^{k-loc}(G)=\langle \gamma _{i}(H)\rangle$ where $H$ varies over all the subgroups of $G$ generated by subsets of size at most $k$.

A group $G$ has $k$-local nilpotency class (at most) $c$ if and only if the subgroup $\gamma _{c+1}^{k-loc}(G)$ is the trivial subgroup.

Particular cases

• 2-local lower central series
• 3-local lower central series