Supplements of bounded permutation groups

Abstract

AbstractLet λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group Bλ(Ω), or simply Bλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of Bλ if BλG is 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 group G acting on the set Ω is a supplement of Bλ if and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliser G{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 developed pcf theory.