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

by Shelah and Steprans. [ShSr:786]

Fundamenta Math, 2007

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.

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