{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\n$G$ is a *one-relator group* if it has a [[presentation|0JGEQXQ4AC]] $\\langle S \\mid r \\rangle$ with a single relator $r$.\nHere $r$ is an element of the [[free group|0KTW26C80J]] $F(S)$ on the set $S$.\nThat is, $G$ is isomorphic to the quotient of $F(S)$ by the normal subgroup generated by $r$.\n\n## Remarks\n\nThe relator may be taken cyclically reduced, since a cyclic permutation of $r$ is a conjugate of $r$ and generates the same normal subgroup.\nA one-relator group has an element of finite order other than $1$ if and only if $r$ is a proper power $s^n$ in $F(S)$ with $n \\ge 2$.\nIn that case, if $s$ is chosen not to be a proper power itself, then $s$ has order exactly $n$ in $G$.\n\nExample: $\\Z^2$ is a one-relator group.\nThe map $\\langle a, b \\mid a b a^{-1} b^{-1} \\rangle \\to \\Z^2$ with $a \\mapsto (1, 0)$ and $b \\mapsto (0, 1)$ is an isomorphism, since the relator says exactly that $a$ and $b$ commute.\n\nNon-example: $\\Z/2 \\times \\Z/2$ is not a one-relator group.\nIf $\\langle S \\mid r \\rangle$ with $S$ finite of size $k$ defined it, its abelianization would be $\\Z^k$ modulo the subgroup generated by one vector, which is $\\Z^{k-1} \\oplus \\Z/d$ for some integer $d \\ge 0$.\nThe group $\\Z/2 \\times \\Z/2$ is abelian, so it equals its abelianization.\nIt is finite, so $k - 1 = 0$, and then it would be cyclic, which it is not.\nA presentation with $S$ infinite gives an abelianization with infinitely many free factors $\\Z$, so it fails as well.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "68AFPHR4GN",
    "kind": "definition",
    "msc": [
      "20F05",
      "20F06"
    ],
    "name": "One-relator group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/One-relator_group"
      }
    ],
    "tag": "PZ9CPKHXVZ",
    "uses": [
      "G5S1M9",
      "0JGEQXQ4AC",
      "0KTW26C80J"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PZ9CPKHXVZ/",
  "links": [
    {
      "field": "defines",
      "target": "68AFPHR4GN"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0JGEQXQ4AC"
    },
    {
      "field": "uses",
      "target": "0KTW26C80J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PZ9CPKHXVZ.md",
  "msc": [
    "20F05",
    "20F06"
  ],
  "name": "One-relator group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PZ9/PZ9CPKHXVZ.md",
  "status": null,
  "tag": "PZ9CPKHXVZ",
  "uri": "m2://PZ9CPKHXVZ",
  "url": "https://math2.tamnd.com/wiki/one-relator-group-definition/",
  "writing": {
    "tag": "PZ9CPKHXVZ",
    "kind": "definition",
    "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
      }
    ]
  }
}