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  "body": "\n## Statement\n\nLet *G* be a group acting on a set *X*. Let *a*\u003csub\u003e1\u003c/sub\u003e, ...,*a*\u003csub\u003e*k*\u003c/sub\u003e be elements of *G* of infinite [[order|0B85S2V2DR]], where *k* ≥ 2. Suppose there exist disjoint nonempty subsets\n\nof $X$ with the following properties:\n\n- $a_{i}(X - X_{i}^{-}) \\subseteq X_{i}^{+}$ for $i = 1, ..., k$;\n- $a_{i}^{-1}(X - X_{i}^{+}) \\subseteq X_{i}^{-}$ for $i = 1, ..., k$.\n\nThen the subgroup $H = \\langle a_{1}, ..., a_{k}\\rangle \\le G$ [[generated|0MBV2VHMXA]] by *a*\u003csub\u003e1\u003c/sub\u003e, ..., *a*\u003csub\u003e*k*\u003c/sub\u003e is [[free|020Z5SEN27]] with free basis $\\{a_{1}, ..., a_{k}\\}$.\n",
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    "kind": "statement",
    "name": "The Ping-pong lemma for infinite cyclic subgroups",
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        "locator": "Ping-pong lemma, section Formal statements \u003e The Ping-pong lemma for infinite cyclic subgroups",
        "mention": "wikipedia://en/18786361#FMBI",
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  "name": "The Ping-pong lemma for infinite cyclic subgroups",
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  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J1P/J1PF2Q7KG1.md",
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  "tag": "J1PF2Q7KG1",
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  "url": "https://math2.tamnd.com/wiki/the-ping-pong-lemma-for-infinite-cyclic-subgroups/",
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