{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet *g* be an element of *G* = π\u003csub\u003e1\u003c/sub\u003e(**A**, *T*) represented as a product of the form\n\n$$g=a_0e_1a_1\\dots e_na_n,$$\n\nwhere *e*\u003csub\u003e1\u003c/sub\u003e, ..., *e\u003csub\u003en\u003c/sub\u003e* is a closed edge-path in *A* with the vertex sequence *v*\u003csub\u003e0\u003c/sub\u003e, *v*\u003csub\u003e1\u003c/sub\u003e, ..., *v\u003csub\u003en\u003c/sub\u003e* = *v*\u003csub\u003e0\u003c/sub\u003e (that is *v*\u003csub\u003e0\u003c/sub\u003e=*o*(*e*\u003csub\u003e1\u003c/sub\u003e), *v\u003csub\u003en\u003c/sub\u003e* = *t*(*e\u003csub\u003en\u003c/sub\u003e*) and *v\u003csub\u003ei\u003c/sub\u003e* = *t*(*e\u003csub\u003ei\u003c/sub\u003e*) = *o*(*e*\u003csub\u003e*i*+1\u003c/sub\u003e) for 0 \\\u003c *i* \\\u003c *n*) and where $a_i\\in A_{v_i}$ for *i* = 0, ..., *n*.\n\nSuppose that *g* = 1 in *G*. Then\n\n- either *n* = 0 and *a*\u003csub\u003e0\u003c/sub\u003e = 1 in $A_{v_0}$,\n- or *n* \u003e 0 and there is some 0 \\\u003c *i* \\\u003c *n* such that *e*\u003csub\u003e*i*+1\u003c/sub\u003e = $\\overline{e_{i}}$ and $a_i\\in \\omega_{e_i}(A_{e_i})$.\n\nThe normal forms theorem immediately implies that the canonical homomorphisms *A\u003csub\u003ev\u003c/sub\u003e* → π\u003csub\u003e1\u003c/sub\u003e(**A**, *T*) are injective, so that we can think of the vertex groups *A*\u003csub\u003e*v*\u003c/sub\u003e as subgroups of *G*.\n\nHiggins has given a nice version of the normal form using the fundamental groupoid of a graph of groups. This avoids choosing a base point or tree, and has been exploited by Moore.\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "inventory": [
      "QS511NKYGG"
    ],
    "kind": "statement",
    "name": "The normal forms theorem",
    "role": "theorem",
    "sources": [
      {
        "locator": "Bass–Serre theory, section Basic set-up \u003e The normal forms theorem",
        "mention": "wikipedia://en/18673708#0P0V",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:18673708@r1364033949"
      }
    ],
    "tag": "MSEHTZY6XZ"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/MSEHTZY6XZ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/MSEHTZY6XZ.md",
  "msc": null,
  "name": "The normal forms theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MSE/MSEHTZY6XZ.md",
  "status": null,
  "tag": "MSEHTZY6XZ",
  "uri": "m2://MSEHTZY6XZ",
  "url": "https://math2.tamnd.com/wiki/the-normal-forms-theorem/",
  "writing": {
    "tag": "MSEHTZY6XZ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}