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