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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 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 1989 and Winkler in 1993. We show 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 2^{aleph_0}. Our purpose is also 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. We prove 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.