3-local lower central series

Definition
The 3-local lower central series of a group $$G$$ is a descending series defined as follows. The $$i^{th}$$ member, which we will denote as $$\gamma_i^{3-loc}(G)$$ is defined as:

$$\gamma_i^{3-loc}(G) = \langle \gamma_i(H) \rangle$$ where $$H$$ varies over all subgroups of $$G$$ that are generated by at most 3 elements and $$\gamma_i(H)$$ denotes the $$i^{th}$$ member of the lower central series of $$H$$.

$$G$$ has 3-local nilpotency class (at most) $$c$$ if and only if $$\gamma_{c + 1}^{3-loc}(G)$$ is the trivial subgroup.

Related notions

 * Local lower central series
 * 2-local lower central series