{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a [[finitely generated group|082DWWXMRJ]].\nCall $G$ *recursively presented* if it has a [[presentation|0JGEQXQ4AC]] $\\langle S \\mid R \\rangle$ with $S$ finite and $R$ [[computably enumerable|TN9G24VABN]].\nHere $R$ is a set of words in the elements of $S$ and their inverses.\nThen $G$ is isomorphic to a [[subgroup|0X1JXT47GA]] of a [[finitely presented group|TJKANTKASZ]] if and only if $G$ is recursively presented.\n\n## Remarks\n\nOne direction is easy.\nLet $G$ be generated by finitely many elements of a finitely presented group $P$.\nA program can list all consequences of the finitely many relators of $P$.\nSo it can list all words in the generators of $G$ that equal $1$ in $P$.\nThe other direction is the hard part of Higman's 1961 proof.\nIt builds the finitely presented group by a long chain of [[HNN extensions|YEYERRY37K]] and amalgamated free products that encode the enumeration of $R$.\n\nExample: the wreath product $\\Z \\wr \\Z$ has the presentation $\\langle a, t \\mid [a, t^n a t^{-n}] = 1 \\text{ for } n \\ge 1 \\rangle$.\nThe set of relators is computably enumerable, so $\\Z \\wr \\Z$ embeds in a finitely presented group, although it is not finitely presented itself.\n\nThere are uncountably many $2$-generated groups up to isomorphism, but only countably many finitely presented groups, each with countably many finitely generated subgroups.\nSo the hypothesis cannot be dropped: some finitely generated groups do not embed in any finitely presented group.\n\n## Explanations\n\n### L1 human\n\nA finitely presented group is described by finitely many generators and finitely many rules.\nThe theorem says exactly which finitely generated groups live inside such groups.\nThey are the groups whose rules can be listed one by one by a computer program, even if the list never ends.\nIt is a bridge between group theory and the theory of computation.\n\n### L2 human\n\nThe theorem shows that there is a finitely presented group that contains every finitely presented group as a subgroup.\nIt also gives another route to finitely presented groups with unsolvable [[word problem|PBGTTAVMMT]].\nTake a computably enumerable set of integers that is not computable.\nEncode it into the relators of a recursively presented group with unsolvable word problem, and embed this group in a finitely presented group.\nThe word problem of the big group cannot be solvable, since it would solve the word problem of the subgroup.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "TJKANTKASZ",
      "082DWWXMRJ"
    ],
    "kind": "statement",
    "msc": [
      "20F10",
      "20F05",
      "03D40"
    ],
    "name": "Higman embedding theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q17029787",
      "wikipedia": "Higman's embedding theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Higman%27s_embedding_theorem"
      }
    ],
    "tag": "P7TZ694FCM",
    "uses": [
      "082DWWXMRJ",
      "TJKANTKASZ",
      "0JGEQXQ4AC",
      "0X1JXT47GA"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/P7TZ694FCM/",
  "links": [
    {
      "field": "about",
      "target": "TJKANTKASZ"
    },
    {
      "field": "about",
      "target": "082DWWXMRJ"
    },
    {
      "field": "uses",
      "target": "082DWWXMRJ"
    },
    {
      "field": "uses",
      "target": "TJKANTKASZ"
    },
    {
      "field": "uses",
      "target": "0JGEQXQ4AC"
    },
    {
      "field": "uses",
      "target": "0X1JXT47GA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/P7TZ694FCM.md",
  "msc": [
    "20F10",
    "20F05",
    "03D40"
  ],
  "name": "Higman embedding theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P7T/P7TZ694FCM.md",
  "status": null,
  "tag": "P7TZ694FCM",
  "uri": "m2://P7TZ694FCM",
  "url": "https://math2.tamnd.com/wiki/higman-embedding-theorem/",
  "writing": {
    "tag": "P7TZ694FCM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}