---
tag: K71BN4C02T
kind: statement
role: lemma
name: Ping-pong lemma for several subgroups
sources:
  - wikipedia: en:18786361@r1361203357
    locator: Ping-pong lemma, section Formal statements > Ping-pong lemma for several subgroups
    note: adopted
    mention: wikipedia://en/18786361#GPNP
    sections: [Statement]
inventory: [1WXV4961FG]
---

## Statement

This 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).

Let *G* be a group acting on a set *X* and let *H*<sub>1</sub>, *H*<sub>2</sub>, ..., *H*<sub>*k*</sub> 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:

- For any $i \ne s$ and for any $h$ in $H_{i}$, $h \ne 1$ we have $h(X_{s}) \subseteq X_{i}$.

Then

$$\langle H_1,\dots, H_k\rangle=H_1\ast\dots \ast H_k.$$