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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.\nThe centraliser of $S$ in $G$ is the set $C_G(S) = \\{g \\in G : gs = sg \\text{ for all } s \\in S\\}$.\nFor an element $a \\in G$, the notation $C_G(a)$ means $C_G(\\{a\\})$.\n\n## Remarks\n\nThe centraliser $C_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$.\nThe centraliser of $G$ itself is the [[centre|0DZ93493VN]] $Z(G)$.\nThe element $a$ lies in its own centraliser $C_G(a)$, and so does each power of $a$.\n\n## Explanations\n\n### L1 human\n\nThe centraliser of an element is the set of elements that commute with it.\nIf the element commutes with everything, its centraliser is the whole group.\nThe size of the centraliser tells you how many conjugates the element has.\n",
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