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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe centre of $G$ is the set $Z(G) = \\{z \\in G : zg = gz \\text{ for all } g \\in G\\}$.\n\n## Remarks\n\nThe centre is an [[abelian|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] of $G$, and it is [[characteristic|0KABPMRHM2]].\nThe group $G$ is abelian if and only if $Z(G) = G$.\nThe centre of the symmetric group $S_n$ is trivial for $n \\geq 3$.\nThe centre of $\\mathrm{GL}_n(\\mathbb{R})$ is the set of the nonzero scalar matrices $\\lambda I$.\n\n## Explanations\n\n### L1 human\n\nThe centre collects the elements that commute with everybody.\nThe identity is always in the centre.\nIn an abelian group the centre is everything, and in many nonabelian groups it is only the identity.\n",
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