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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe outer automorphism group of $G$ is the [[quotient group|0KYX2N69J0]] $\\operatorname{Out}(G) = \\operatorname{Aut}(G) / \\operatorname{Inn}(G)$.\n\n## Remarks\n\nThe quotient exists because $\\operatorname{Inn}(G)$ is a normal subgroup of $\\operatorname{Aut}(G)$.\nAn automorphism that is not inner is called an outer automorphism, but the elements of $\\operatorname{Out}(G)$ are cosets, not automorphisms.\nFor an abelian group, $\\operatorname{Inn}(G)$ is trivial, so $\\operatorname{Out}(G) \\cong \\operatorname{Aut}(G)$.\nThe symmetric group $S_6$ is the only symmetric group $S_n$ with nontrivial outer automorphism group.\n\n## Explanations\n\n### L1 human\n\nSome symmetries of a group come from conjugation by its own elements, and some do not.\nThe outer automorphism group keeps only the second kind, up to the first kind.\nFor many groups it is small or trivial.\n",
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