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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe commutator subgroup of $G$ is the [[subgroup generated by|0RP7VWQ0YN]] the set of all [[commutators|0PXW0BN7DW]] $[g, h]$ with $g, h \\in G$.\nIt is written $[G, G]$ or $G'$.\n\n## Remarks\n\nA product of commutators need not be a commutator, so in general the set of commutators is not a subgroup.\nThe subgroup $[G, G]$ is [[characteristic|0KABPMRHM2]], so it is normal in $G$.\nThe group $G$ is abelian if and only if $[G, G] = \\{e\\}$.\nThe commutator subgroup of the symmetric group $S_n$ is the alternating group $A_n$.\n\n## Explanations\n\n### L1 human\n\nThe commutator subgroup collects all the failures of commutativity in a group.\nIf you divide the group by it, you get an abelian group.\nA large commutator subgroup means a group that is far from abelian.\n",
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