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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.\nThe normaliser of $S$ in $G$ is the set $N_G(S) = \\{g \\in G : gSg^{-1} = S\\}$, where $gSg^{-1} = \\{gsg^{-1} : s \\in S\\}$.\n\n## Remarks\n\nThe normaliser $N_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$, and it contains the [[centraliser|02YYV36BFM]] $C_G(S)$.\nIf $H$ is a subgroup of $G$, then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $N_G(H)$. Also, $N_G(H)$ is the largest subgroup of $G$ with this property.\nThus $H$ is normal in $G$ exactly when $N_G(H) = G$.\nThe number of [[conjugates|0H11ZPX4ND]] of $H$ in $G$ is the index $[G : N_G(H)]$.\n\n## Explanations\n\n### L1 human\n\nThe normaliser of a subgroup is the set of elements that do not move the subgroup when they conjugate it.\nThey may move the elements of the subgroup around, but the subgroup as a set stays in place.\nThe normaliser is big when the subgroup is close to normal.\n",
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