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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \\in G$.\nThe conjugate of $H$ by $g$ is the set $gHg^{-1} = \\{ghg^{-1} : h \\in H\\}$.\nTwo subgroups $H$ and $K$ of $G$ are conjugate when $K = gHg^{-1}$ for some $g \\in G$.\n\n## Remarks\n\nThe set $gHg^{-1}$ is a subgroup of $G$, and $h \\mapsto ghg^{-1}$ is an isomorphism from $H$ to $gHg^{-1}$.\nConjugacy of subgroups is an equivalence relation on the set of subgroups of $G$.\nA subgroup that equals each of its conjugates is a normal subgroup.\n\n## Explanations\n\n### L1 human\n\nConjugating a subgroup moves it to another position inside the group, but keeps its shape.\nIn the symmetric group $S_3$, the three subgroups of order $2$ are conjugate to each other.\nConjugate subgroups have the same order and the same structure.\n",
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  "name": "Conjugate subgroup",
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