Abstract

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation groupBλ(Ω), or simplyBλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation groupGacting on Ω is asupplement of BλifBλGis the full symmetric group on Ω.In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a groupGacting on the set Ω is a supplement ofBλif and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliserG{Δ}acts as the full symmetric group on Ω ∖ Δ. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developedpcftheory.