Parker Vectors for Infinite Groups

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that calculating Parker vectors of oligomorphic groups is akin to counting circulant combinatorial objects, mirroring in a sense the combinatorial meaning of the orbit-counting sequence of an oligomorphic group. Finally we explicitly find the Parker vectors for some groups, one of which being the automorphism group of the Rado graph.