= Ping-pong lemma
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= Klein combination theorem
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= Klein's combination theorem
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Suppose nontrivial <groups> $G_1,G_2$ act on a <set> and there are disjoint nonempty <subsets> $X_1,X_2$ such that every nonidentity element of $G_1$ maps $X_2$ into $X_1$, while every nonidentity element of $G_2$ maps $X_1$ into $X_2$. Assume at least one factor has at least three elements. Then the subgroup generated by $G_1$ and $G_2$ is their <free product>. The cardinality hypothesis excludes the counterexample in which $G_1=G_2=C_2$ acts by swapping $X_1$ and $X_2$. Versions allowing both factors to have order two need an additional separation hypothesis.
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