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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nAn automorphism of $G$ is a [[group isomorphism|08MG7H8JW5]] from $G$ to $G$.\nThe automorphism group $\\operatorname{Aut}(G)$ is the set of all automorphisms of $G$, with composition of maps as the operation.\n\n## Remarks\n\nThe identity map is the identity element of $\\operatorname{Aut}(G)$, and the inverse of an automorphism is its inverse map.\nThe automorphism group of $(\\mathbb{Z}, +)$ has two elements, the identity and $n \\mapsto -n$.\nFor an [[abelian group|082GHNPJ53]] $G$, the map $g \\mapsto g^{-1}$ is an automorphism.\n\n## Explanations\n\n### L1 human\n\nAn automorphism is a way to rename the elements of a group so that all products stay correct.\nThe set of all such renamings is again a group, because you can compose and undo renamings.\nFor the integers, the only renamings are \"do nothing\" and \"change all signs\".\n",
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