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  "body": "\n## Definition\n\nA [[group|G5S1M9]] $G$ is nilpotent when it has a central series. This is a finite sequence $\\{e\\} = N_0 \\subseteq N_1 \\subseteq \\cdots \\subseteq N_n = G$ of [[normal subgroups|0CB0DX5BVV]] of $G$ with $N_{i+1}/N_i \\subseteq Z(G/N_i)$ for each $i \u003c n$.\nHere $Z(G/N_i)$ is the [[centre|0DZ93493VN]] of the [[quotient group|0KYX2N69J0]] $G/N_i$.\n\n## Remarks\n\nThe condition $N_{i+1}/N_i \\subseteq Z(G/N_i)$ means $[N_{i+1}, G] \\subseteq N_i$.\nEquivalently, the [[lower central series|0K98R7A4VC]] reaches $\\{e\\}$, or the [[upper central series|0XW54DH28V]] reaches $G$; the least such length is the nilpotency class.\nThe nilpotent groups of class at most $1$ are exactly the [[abelian groups|082GHNPJ53]].\nThe quaternion group $Q_8$ is nilpotent of class $2$, but $S_3$ is not nilpotent.\n\n## Explanations\n\n### L1 human\n\nA nilpotent group is close to abelian in a strong sense.\nEach layer of a central series commutes with the whole group, up to the lower layers.\nEach nilpotent group is solvable, but the converse is false, for example for $S_3$.\n",
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