2-local lower central series

Definition

The 2-local lower central series of a group $G$ is a descending series defined as follows. The $i^{th}$ member, which we will denote as $\gamma _{i}^{2-loc}(G)$ is defined as:

$\gamma _{i}^{2-loc}(G)=\langle \gamma _{i}(H)\rangle$ 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 2-local nilpotency class (at most) $c$ if and only if $\gamma _{c+1}^{2-loc}(G)$ is the trivial subgroup.

Related notions

• Local lower central series
• 3-local lower central series