Orbit-equivalent infinite permutation groups

TL;DR

This paper proves that for certain infinite permutation groups, orbit-equivalence combined with primitivity but not 2-transitivity implies the groups are identical, clarifying the structure of such groups.

Contribution

It establishes a new uniqueness result for orbit-equivalent primitive but not 2-transitive infinite permutation groups.

Findings

Orbit-equivalence plus primitivity implies group equality under specified conditions.

Provides a characterization of infinite permutation groups with certain orbit properties.

Clarifies the structure of primitive but not 2-transitive groups in the infinite case.

Abstract

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.