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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThe subgroup $N$ is normal in $G$ when $gng^{-1} \\in N$ for all $g \\in G$ and all $n \\in N$.\nThe notation is $N \\trianglelefteq G$.\n\n## Remarks\n\nEquivalently, $gNg^{-1} = N$ for each $g \\in G$, that is, $N$ equals each of its [[conjugates|0H11ZPX4ND]].\nEach group $G$ has the normal subgroups $\\{e\\}$ and $G$.\nIn the symmetric group $S_3$, the subgroup $\\{e, (1\\,2\\,3), (1\\,3\\,2)\\}$ is normal, but the subgroup $\\{e, (1\\,2)\\}$ is not.\n\n## Explanations\n\n### L1 human\n\nA normal subgroup does not move when you conjugate it by any element of the group.\nThis is the condition that lets you divide the group by the subgroup and get a new group.\nIn an abelian group, every subgroup is normal.\n",
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