---
tag: J1PF2Q7KG1
kind: statement
role: lemma
name: The Ping-pong lemma for infinite cyclic subgroups
sources:
  - wikipedia: en:18786361@r1361203357
    locator: Ping-pong lemma, section Formal statements > The Ping-pong lemma for infinite cyclic subgroups
    note: adopted
    mention: wikipedia://en/18786361#FMBI
    sections: [Statement]
inventory: [Q92JN3J54J]
---

## Statement

Let *G* be a group acting on a set *X*. Let *a*<sub>1</sub>, ...,*a*<sub>*k*</sub> be elements of *G* of infinite [[order|0B85S2V2DR]], where *k* ≥ 2. Suppose there exist disjoint nonempty subsets

of $X$ with the following properties:

- $a_{i}(X - X_{i}^{-}) \subseteq X_{i}^{+}$ for $i = 1, ..., k$;
- $a_{i}^{-1}(X - X_{i}^{+}) \subseteq X_{i}^{-}$ for $i = 1, ..., k$.

Then the subgroup $H = \langle a_{1}, ..., a_{k}\rangle \le G$ [[generated|0MBV2VHMXA]] by *a*<sub>1</sub>, ..., *a*<sub>*k*</sub> is [[free|020Z5SEN27]] with free basis $\{a_{1}, ..., a_{k}\}$.