## Abstract

With any permutation group G on an infinite set Ω is associated a graded algebra script A sign^{G} (the algebra of G-invariants in the reduced incidence algebra of finite subsets of Ω). The dimension of the nth homogeneous component of script A sign^{G} is equal to the number of orbits of G on n-subsets of Ω. If it happens that G is the automorphism group of a homogeneous structure M, then this is the number of unlabelled n-element substructures of M. Many combinatorial enumeration problems fit into this framework. I conjectured 20 years ago that, if G has no finite orbits on Ω, then script A sign^{G} is an integral domain (and even has the stronger property that a specific quotient is an integral domain). I shall say that G is entire if script A sign^{G} is an integral domain, and strongly entire if the stronger property holds. These properties have (rather subtle) consequences for the enumeration problems. The conjecture is still open; but in this paper I prove that, if G is transitive on Ω and the point stabilizer H is (strongly) entire, then G is (strongly) entire.

Original language | English |
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Pages (from-to) | 173-179 |

Number of pages | 7 |

Journal | Journal of Group Theory |

Volume | 1 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 1998 |