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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThe normal core of $H$ in $G$ is the subgroup $\\operatorname{Core}_G(H) = \\bigcap_{g \\in G} gHg^{-1}$.\n\n## Remarks\n\nThe core is a [[normal subgroup|0CB0DX5BVV]] of $G$, and it contains each normal subgroup of $G$ that lies in $H$.\nThe subgroup $H$ is normal exactly when $\\operatorname{Core}_G(H) = H$.\nThe core is the kernel of the action of $G$ on the left cosets of $H$ by left multiplication.\n\n## Explanations\n\n### L1 human\n\nSome subgroups are not normal, but they still contain normal subgroups.\nThe normal core is the biggest of these.\nYou find it by keeping only the elements that lie in every conjugate of the subgroup.\n",
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