Geodesic
Orbit-equivalent infinite permutation groups.
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 973-988.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $G=H$.
Classification : 20B07, 20B15, 20B27, 20B35
Keywords: orbit-equivalent permutation groups, closed permutation groups, primitive permutation groups, topologies on symmetric groups 
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     author = {Lockett, D.C. and Macpherson, H.D.},
     title = {Orbit-equivalent infinite permutation groups.},
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Lockett, D.C.; Macpherson, H.D. Orbit-equivalent infinite permutation groups.. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 973-988. http://geodesic.mathdoc.fr/item/JAC_2013__38_4_a1/