Lower central series

Definition

Let $G$ be a group. For subgroups $H$ and $K$ of $G$, let $[H, K]$ be the subgroup generated by the commutators $[h, k]$ with $h \in H$ and $k \in K$. The lower central series of $G$ is the sequence $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$.

Remarks

The second term $\gamma_2(G) = [G, G]$ is the commutator subgroup. Each $\gamma_i(G)$ is normal in $G$, and $\gamma_i(G)/\gamma_{i+1}(G)$ lies in the centre of $G/\gamma_{i+1}(G)$. The terms of the derived series satisfy $G^{(i)} \subseteq \gamma_{i+1}(G)$. For $S_3$, the series is $S_3 \supseteq A_3 \supseteq A_3 \supseteq \cdots$, so it never reaches $\{e\}$.

Explanations

L1 human

The lower central series starts with the whole group. At each step, it takes the commutators of the last term with all elements of the group. If this process reaches the identity, the group is nilpotent.