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  "body": "\n## Statement\n\nFor each positive integer $n$ there is an integer $J(n)$ with the following property.\nEvery finite subgroup $G$ of the [[general linear group|0FCHE902D7]] $\\mathrm{GL}_n(\\C)$ has an [[abelian|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] $A$ of [[index|0DTXBQPA2S]] $[G : A] \\le J(n)$.\n\n## Remarks\n\nJordan proved the theorem for finite groups.\nSchur extended it to [[torsion groups|JZNQTTMG58]] of matrices: every torsion subgroup of $\\mathrm{GL}_n(\\C)$ has an abelian normal subgroup of finite index.\n\nIdea of the proof: a finite group $G$ preserves a Hermitian inner product, so after a change of basis $G$ lies in the [[unitary group|0R9R0Y1HDK]] $U(n)$.\nIf two unitary matrices are close enough to the identity, their commutator is even closer to the identity.\nIn a finite group this forces the elements in a small ball around the identity to generate an abelian subgroup $A$.\nThis $A$ is normal, since conjugation by unitary matrices preserves distances to the identity.\nIts index is at most the number of small balls needed to cover $U(n)$, which depends only on $n$.\n\nExample: for $n = 2$, the dihedral group of order $2m$ acts on $\\C^2$ by rotations and reflections of the plane.\nIt is not abelian for $m \\ge 3$, but its rotations form a cyclic normal subgroup of index $2$.\n\nThe finiteness or torsion hypothesis is necessary.\nThe group $\\mathrm{SL}_2(\\Z) \\subseteq \\mathrm{GL}_2(\\C)$ contains a nonabelian [[free group|0KTW26C80J]] of finite index, and a nonabelian free group has no abelian subgroup of finite index.\n\n## Explanations\n\n### L1 human\n\nFinite groups of invertible matrices of a fixed size are almost abelian.\nEach one contains an abelian piece whose number of cosets is bounded by a number that depends only on the size of the matrices.\nSo a finite group of $3 \\times 3$ matrices can be huge, but most of its complexity is commutative.\n\n### L2 human\n\nThe bound $J(n)$ does not depend on the order of $G$, and this is the whole point.\nThe same commutator argument in the Euclidean group is behind the first of the [[Bieberbach theorems|YE87R5CDJY]] on [[crystallographic groups|7KTJKFQT2N]].\nThe theorem also shows that a finite group with no abelian normal subgroup of small index, such as a large alternating group, has no faithful complex representation of small degree.\n",
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