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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be a [[structure|983GZBQYT4]] for a signature $\\sigma$.\nAn *automorphism* of $\\mathbf{A}$ is an [[isomorphism|PZMGQMDN25]] from $\\mathbf{A}$ to $\\mathbf{A}$.\nThe set of automorphisms of $\\mathbf{A}$ is written $\\Aut(\\mathbf{A})$.\n\n## Remarks\n\nExample: the map $z \\mapsto \\bar{z}$ is an automorphism of the field $(\\C, +, \\cdot, 0, 1)$, because it is a bijection of $\\C$ to itself with $\\overline{z + w} = \\bar{z} + \\bar{w}$, $\\overline{zw} = \\bar{z}\\,\\bar{w}$, $\\bar{0} = 0$ and $\\bar{1} = 1$.\n\nNon-example: the map $x \\mapsto 2x$ on $(\\Z, +, 0)$ is an injective homomorphism but not an automorphism, because it is not surjective: $1$ is not in its image.\n\nThe set $\\Aut(\\mathbf{A})$ is a group under composition, with the identity map as identity element.\nThe only automorphism of the field $\\R$ is the identity.\n",
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