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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.\nThen the [[quotient group|0KYX2N69J0]] $G/N$ is [[abelian|082GHNPJ53]] if and only if $[G, G] \\subseteq N$, where $[G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].\n\n## Remarks\n\nThe cosets $gN$ and $hN$ commute exactly when $g^{-1} h^{-1} g h \\in N$.\nThus $[G, G]$ is the smallest normal subgroup of $G$ with an abelian quotient.\nA subgroup that contains $[G, G]$ is automatically normal.\n\n## Explanations\n\n### L1 human\n\nTo make a group abelian by a quotient, you must send each commutator to the identity.\nSo the subgroup that you divide by must contain all commutators.\nIf it does, the quotient is abelian.\n",
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