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SUPPLEMENTS OF BOUNDED GROUPS STEPHEN BIGELOW
 

Summary: SUPPLEMENTS OF BOUNDED GROUPS
STEPHEN BIGELOW
Abstract. Let be in nite cardinals and let be a set of cardinality . The bounded
permutation group B ( ), or simply B , is the group consisting of all permutations of which
move fewer than points in . We say that a permutationgroup G acting on is a supplement
of B if B G is the full symmetric group on .
In 7], Macpherson and Neumann claimed to have classi ed all supplements of bounded
permutation groups. Speci cally, they claimed to have proved that a group G acting on the set
is a supplement of B if and only if there exists with j j < such that the setwise
stabiliser Gf g acts as the full symmetric group on n . 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 conditionsunder which a counterexamplecan be constructed.
In the process we will discover surprising links with cardinal arithmetic and Shelah's recently
developed pcf theory.
1. Introduction
This paper concerns an error in Macpherson and Neumann's paper \Subgroups of In nite Sym-
metric Groups" 7]. The proof they give for their Theorem 1.2 contains a tacit assumption that
all cardinals are regular. In this paper I will demonstrate conditions under which the theorem is
true as well as conditions under which a counterexample can be constructed.
Suppose ? is an in nite set. We denote the full symmetric group on ? by Sym(?). If j?j

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics