Supplements of Bounded Permutation Groups

Let ). < i, be infinite cardinals and let Q be a set of cardinality 'c. The bounded permutation group B. (Q), or simply B2, is the group consisting of all permutations of Q which move fewer than A points in Q. We say that a permutation group G acting on Q is a supplement of B2 if BA G is the full symmetric group on Q. 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 Q is a supplement of B. if and only if there exists A C Q with JAI < 2 such that the setwise stabiliser G{A} acts as the full symmetric group on Q \ A. However I have found a mistake in their proof. The aim of this paper is to examine conditions tinder which Macpherson and Neumann's claim holds, as well as conditions tinder which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory. ?