{
  "aliases": null,
  "backlinks": [
    {
      "field": "proves",
      "target": "R9XA44DYC5"
    }
  ],
  "body": "\n## Statement\n\nThis version of the ping-pong lemma ensures that several subgroups of a group [[acting|0WCY42EV9V]] on a set generate a free product. The following statement appears in Olijnyk and Suchchansky (2004), and the proof is from de la Harpe (2000).\n\nLet *G* be a group acting on a set *X* and let *H*\u003csub\u003e1\u003c/sub\u003e, *H*\u003csub\u003e2\u003c/sub\u003e, ..., *H*\u003csub\u003e*k*\u003c/sub\u003e be subgroups of *G* where *k* ≥ 2, such that at least one of these subgroups has [[order|0B85S2V2DR]] greater than 2. Suppose there exist [[pairwise disjoint|0W4RPA8QH3]] [[nonempty|0ECVK59Y03]] subsets $X_{1}, X_{2}, ...,X_{k}$ of $X$ such that the following holds:\n\n- For any $i \\ne s$ and for any $h$ in $H_{i}$, $h \\ne 1$ we have $h(X_{s}) \\subseteq X_{i}$.\n\nThen\n\n$$\\langle H_1,\\dots, H_k\\rangle=H_1\\ast\\dots \\ast H_k.$$\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "inventory": [
      "1WXV4961FG"
    ],
    "kind": "statement",
    "name": "Ping-pong lemma for several subgroups",
    "role": "lemma",
    "sources": [
      {
        "locator": "Ping-pong lemma, section Formal statements \u003e Ping-pong lemma for several subgroups",
        "mention": "wikipedia://en/18786361#GPNP",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:18786361@r1361203357"
      }
    ],
    "tag": "K71BN4C02T"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/K71BN4C02T/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/K71BN4C02T.md",
  "msc": null,
  "name": "Ping-pong lemma for several subgroups",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K71/K71BN4C02T.md",
  "status": null,
  "tag": "K71BN4C02T",
  "uri": "m2://K71BN4C02T",
  "url": "https://math2.tamnd.com/wiki/ping-pong-lemma-for-several-subgroups/",
  "writing": {
    "tag": "K71BN4C02T",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}