{
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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nFor subgroups $H$ and $K$ of $G$, let $[H, K]$ be the subgroup [[generated by|0RP7VWQ0YN]] the [[commutators|0PXW0BN7DW]] $[h, k]$ with $h \\in H$ and $k \\in K$.\nThe lower central series of $G$ is the sequence $\\gamma_1(G) = G$ and $\\gamma_{i+1}(G) = [\\gamma_i(G), G]$ for $i \\geq 1$.\n\n## Remarks\n\nThe second term $\\gamma_2(G) = [G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].\nEach $\\gamma_i(G)$ is normal in $G$, and $\\gamma_i(G)/\\gamma_{i+1}(G)$ lies in the centre of $G/\\gamma_{i+1}(G)$.\nThe terms of the [[derived series|0Q5G5DJ4P2]] satisfy $G^{(i)} \\subseteq \\gamma_{i+1}(G)$.\nFor $S_3$, the series is $S_3 \\supseteq A_3 \\supseteq A_3 \\supseteq \\cdots$, so it never reaches $\\{e\\}$.\n\n## Explanations\n\n### L1 human\n\nThe lower central series starts with the whole group.\nAt each step, it takes the commutators of the last term with all elements of the group.\nIf this process reaches the identity, the group is nilpotent.\n",
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