{
  "aliases": null,
  "backlinks": [
    {
      "field": "instance_of",
      "target": "429JJ5ZNBY"
    },
    {
      "field": "about",
      "target": "GWZN3EKEWD"
    },
    {
      "field": "uses",
      "target": "GWZN3EKEWD"
    },
    {
      "field": "defines",
      "target": "TACX13XZST"
    }
  ],
  "body": "\n## Summary\n\nAn HNN extension of a [[group|G5S1M9]] $G$ adds one new element $t$.\nThis element conjugates a [[subgroup|0X1JXT47GA]] $H$ of $G$ onto another subgroup $K$ by a given [[isomorphism|08MG7H8JW5]].\nThe group $G$ embeds in the new group, so two isomorphic subgroups of $G$ become conjugate in a larger group.\nThe construction is named after Graham Higman, Bernhard Neumann and Hanna Neumann.\nTogether with the [[free product|TM7QS7P7X9]] with amalgamation, it is a basic way to build groups from smaller ones.\nIt is used in the proof of the [[Higman embedding theorem|P7TZ694FCM]].\n\n## Explanations\n\n### L1 human\n\nStart with a group $G$ in which two subgroups $H$ and $K$ look alike, through an isomorphism $\\alpha$ from $H$ to $K$.\nInside $G$ there may be no element that turns $H$ into $K$ by conjugation.\nThe HNN extension adds a new letter $t$ and the rules $t h t^{-1} = \\alpha(h)$, and nothing else.\nFor example, take $G = \\Z$ written as the powers of $a$, take $H = \\langle a \\rangle$, $K = \\langle a^2 \\rangle$ and $\\alpha(a) = a^2$.\nThe HNN extension is $\\langle a, t \\mid t a t^{-1} = a^2 \\rangle$, the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(1, 2)$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "20E06",
      "20F05"
    ],
    "name": "HNN extension",
    "same_as": {
      "wikidata": "Q3062521",
      "wikipedia": "HNN extension"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/HNN_extension"
      }
    ],
    "tag": "YEYERRY37K"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/YEYERRY37K/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/YEYERRY37K.md",
  "msc": [
    "20E06",
    "20F05"
  ],
  "name": "HNN extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YEY/YEYERRY37K.md",
  "status": null,
  "tag": "YEYERRY37K",
  "uri": "m2://YEYERRY37K",
  "url": "https://math2.tamnd.com/wiki/hnn-extension/",
  "writing": {
    "tag": "YEYERRY37K",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}