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P.J.Cameron introduced the orbit algebra of a permutation group andconjectured that this algebra is an integral domain if and only if the grouphas no finite orbit. We prove that this conjecture holds and in fact that theage algebra of a relational structure $R$ is an integral domain if and only if$R$ is age-inexhaustible. We deduce these results from a combinatorial lemmaasserting that if a product of two non-zero elements of a set algebra is zerothen there is a finite common tranversal of their supports. The proof is builton Ramsey theorem and the integrity of a shuffle algebra.

When the orbit algebra of group is an integral domain? Proof of a conjecture of P.J. Cameron