{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "W0KBNZBCRR"
    },
    {
      "field": "defines",
      "target": "XY6CJMDY5M"
    }
  ],
  "body": "\n## Summary\n\nAn automorphism of a [[structure|983GZBQYT4]] is an [[isomorphism|PZMGQMDN25]] from the structure to itself.\nIt is a symmetry of the structure: a permutation of its elements that keeps all operations and relations.\nThe automorphisms of a structure form a group under composition, the [[automorphism group|0K1SCPBNKJ]].\nComplex conjugation, as an automorphism of the field $\\C$, and the [[Frobenius automorphism of a finite field|0CGCPX8Z70]] are classical examples.\n\n## Explanations\n\n### L1 human\n\nAn automorphism is a way to shuffle the elements of a structure so that nobody can tell the difference afterwards.\nFor the integers with addition, the map $x \\mapsto -x$ is an automorphism: it turns sums into sums and $0$ into $0$.\nFor a square drawn as a graph with four corners and four sides, each rotation and reflection of the square gives an automorphism.\nThe identity map is always an automorphism.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08A35",
      "03C07"
    ],
    "name": "Automorphism",
    "same_as": {
      "wikidata": "Q782566",
      "wikipedia": "Automorphism"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Automorphism"
      }
    ],
    "specializes": [
      "PZMGQMDN25"
    ],
    "tag": "DSQ17TNW1C"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DSQ17TNW1C/",
  "links": [
    {
      "field": "specializes",
      "target": "PZMGQMDN25"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DSQ17TNW1C.md",
  "msc": [
    "08A35",
    "03C07"
  ],
  "name": "Automorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DSQ/DSQ17TNW1C.md",
  "status": null,
  "tag": "DSQ17TNW1C",
  "uri": "m2://DSQ17TNW1C",
  "url": "https://tamnd.github.io/math2/wiki/automorphism/",
  "writing": {
    "tag": "DSQ17TNW1C",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}