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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $[G, G]$ be its [[commutator subgroup|0ASGSTXSBW]].\nThe abelianisation of $G$ is the [[quotient group|0KYX2N69J0]] $G^{\\mathrm{ab}} = G/[G, G]$.\n\n## Remarks\n\nThe group $G^{\\mathrm{ab}}$ is [[abelian|082GHNPJ53]], by [[this criterion|0MS8QGA6S5]].\nIf $G$ is abelian, then $G^{\\mathrm{ab}} = G$.\nThe abelianisation of the symmetric group $S_n$ for $n \\geq 2$ is cyclic of order $2$, and the sign of a permutation gives the isomorphism.\n\n## Explanations\n\n### L1 human\n\nThe abelianisation forces all elements of a group to commute, and changes nothing more than it must.\nIt keeps as much information about the group as an abelian group can keep.\nFor a group that is already abelian, it does nothing.\n",
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