Possible Cardinalities of Maximal Abelian Subgroups of Quotients of Permutation Groups of the Integers

The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski [8, 7] has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and, furthermore, all the maximal Abelian subgroups are conjugate. For finite symmetric groups the question of the size of maximal Abelian subgroups has been examined by Burns and Goldsmith in [4] and Winkler in [15]. It will be shown in Corollary 3.1 that there is not much interest in generalizing this study to infinite symmetric groups; the cardinality of any maximal Abelian subgroup of the symmetric group of the integers is 20 . The purpose of this paper is to examine the size of maximal Abelian subgroups for a class of groups closely related to the the symmetric group of the integers; these arise by taking an ideal on the integers, considering the subgroup of all permutations which respect the ideal and then taking the quotient by the normal subgroup of permutations which fix all integers except a set in the ideal. It will be shown that the maximal size of Abelian subgroups in such groups is sensitive to the nature of the ideal as well as various set theoretic hypotheses. The reader familiar with applications of the Axiom of Choice may not be surprised by the assertion just made since, one can imagine constructing ideals on the integers by transfinite induction such that the quotient group just described exhibits various desired properties. Consequently, it is of interest to restrict attention to only those ideals which do not require the Axiom of Choice for their definition. All of the ideals considered will here will have simple definitions — indeed, they will all be Borel subsets of P(ω) with the usual topology — and, in fact, the first three sections will focus on the ideal of finite sets. It should be mentioned that there is large body of work examining the analogous quotients of the Boolean algebra P(ω) modulo an analytic ideal — the monograph [5] by Farah is a good reference for this subject. However, the analogy is far from perfect since, for example, whereas the Boolean algebra P(ω)/[ω]0 can consistently have 2 א0 automorphisms [9] it is shown in [1] that the quotient of the full symmetric group of the integers modulo the subgroup of finite permutations has only countably many outer automorphisms. Nevertheless, it may be possible to employ methods similar to those of [5] in order to distinguish between different quotient algebras up to isomorphism. This has been done for elementary equivalence in [11, 13] for quotients of the full symmetric group on κ by the normal subgroups fixing all but λ elements. However since the full symmetric group of the integers has only two proper normal subgroups [10] quotients of certain naturally arising subgroups will be considered instead. One of the goals of this study is to use the cardinal invariant associated with maximal Abelian subgroups as a tool to distinguish between isomorphism types of such groups. In order to state the main results precisely some notation is needed.