# Ping-pong lemma

From Groupprops

## Contents

## Statement

### Statement in terms of two elements and free groups

Suppose is a group with a group action on a set . Further, suppose and are two subsets of such that neither is contained in the other. Further, suppose are elements such that:

- .
- .

Then, the subgroup is a free group with as a freely generating set.

### Statement in terms of two subgroups and free products

Suppose is a group with a group action on a set . Further, suppose and are two subsets of such that neither is contained in the other. Further, suppose are subgroups such that:

- .
- .

Then, the subgroup is an internal free product of the subgroups and .