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  "body": "\n## Description\n\nLet $n \\ge 1$ and let $F_n$ be the [[free group|0KTW26C80J]] of rank $n$.\nThe group $\\mathrm{Out}(F_n) = \\Aut(F_n) / \\mathrm{Inn}(F_n)$ is the [[outer automorphism group|0Q8Y862DTZ]] of $F_n$.\nIt is the quotient of the [[automorphism group|0K1SCPBNKJ]] of $F_n$ by its normal subgroup of inner automorphisms $x \\mapsto g x g^{-1}$.\n\n## Remarks\n\nFor $n = 1$, $F_1 = \\Z$ has the two automorphisms $x \\mapsto \\pm x$ and no nontrivial inner automorphisms, so $\\mathrm{Out}(F_1) \\cong \\Z/2\\Z$.\nEvery automorphism of $F_n$ induces an automorphism of the abelianisation $\\Z^n$, and inner automorphisms induce the identity.\nThis gives a homomorphism $\\mathrm{Out}(F_n) \\to \\mathrm{GL}_n(\\Z)$, into the [[general linear group|0FCHE902D7]] over $\\Z$, which is surjective, because elementary matrices and sign changes lift to automorphisms of $F_n$.\nFor $n = 2$ this map is an isomorphism, so $\\mathrm{Out}(F_2) \\cong \\mathrm{GL}_2(\\Z)$.\nFor $n \\ge 3$ its kernel is infinite.\n\n$\\mathrm{Out}(F_n)$ is finitely presented.\nCuller and Vogtmann built a contractible space, Outer space, on which $\\mathrm{Out}(F_n)$ acts properly; it plays the role that Teichmüller space plays for the [[mapping class group|JV5NYG913V]] of a surface.\n$\\mathrm{Out}(F_n)$ satisfies the [[Tits alternative|QYXGVES1XM]].\n\n## Explanations\n\n### L1 human\n\nThe free group on $n$ letters has many symmetries: for example, one may swap two letters, invert a letter, or replace $a$ by $ab$.\nSome symmetries are boring, namely conjugation by a fixed element; dividing them out gives $\\mathrm{Out}(F_n)$.\nIt is the free-group analogue of the group $\\mathrm{GL}_n(\\Z)$ of integer matrices with integer inverses, and for $n = 2$ the two groups coincide.\nIt is a central object of [[geometric group theory|2T3218RXTD]].\n",
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