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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $a \\in G$.\nThe conjugacy class of $a$ is the set $\\operatorname{Cl}(a) = \\{gag^{-1} : g \\in G\\}$.\n\n## Remarks\n\nThe conjugacy classes are the equivalence classes of the relation \"is [[conjugate to|0DXD4QSZ8A]]\", so they partition $G$.\nThey are the [[orbits|0YQPZB5Q1G]] of the [[action by conjugation|0P8CY9V437]].\nThe class of $a$ is $\\{a\\}$ if and only if $a$ lies in the [[centre|0DZ93493VN]] of $G$.\nIn the symmetric group $S_n$, the conjugacy classes are given by the [[cycle types|0J438GAC8K]].\n\n## Explanations\n\n### L1 human\n\nThe conjugacy class of an element collects all the elements that look like it after a renaming by conjugation.\nIn an abelian group, each element is alone in its class.\nIn $S_3$, the three transpositions form one class, and the two $3$-cycles form another class.\n",
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