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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe upper central series of $G$ is the sequence of normal subgroups $Z_0 = \\{e\\}$ and $Z_{i+1} = \\{x \\in G : [x, y] \\in Z_i \\text{ for all } y \\in G\\}$ for $i \\geq 0$.\n\n## Remarks\n\nThe first term $Z_1$ is the [[centre|0DZ93493VN]] $Z(G)$, and $Z_2$ is called the second centre.\nFor each $i$, $Z_{i+1}/Z_i$ is the centre of the [[quotient group|0KYX2N69J0]] $G/Z_i$.\nFor the quaternion group $Q_8$, the series is $\\{1\\} \\subseteq \\{1, -1\\} \\subseteq Q_8$.\nA group with trivial centre, such as $S_3$, has $Z_i = \\{e\\}$ for each $i$.\n\n## Explanations\n\n### L1 human\n\nThe upper central series starts with the identity and first adds the centre.\nThen it adds the elements that are central modulo the centre, and so on.\nIf this process reaches the whole group, the group is nilpotent.\n",
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