Parker vectors for infinite groups (Q5952152)

Let \(G\) be a permutation group acting on a finite or countable set. If the number \(p_i(G)\) of orbits of \(G\) on \(i\)-cycles is finite for each \(i\in\mathbb{N}\), then the sequence \({\mathbf p}(G)=(p_1(G),p_2(G),p_3(G),\dots)\) is called the Parker vector of the group \(G\).NEWLINENEWLINENEWLINEIt is shown that many results for finite groups extend naturally to the infinite (oligomorphic) case under suitable assumptions. In addition, an interesting connection between Parker vectors for oligomorphic groups and finite circulant substructures in homogeneous relational structures is worked out.NEWLINENEWLINENEWLINEFor some groups explicit Parker vectors are determined, e.g. for the automorphism group of the Rado graph (countable random graph).