Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers

Saharon Shelah ¹   ; Juris Steprāns ²

¹ Department of Mathematics Rutgers University Hill Center, Piscataway NJ 08854-8019, U.S.A. and Institute of Mathematics Hebrew University Givat Ram, Jerusalem 91904, Israel
² Department of Mathematics York University 4700 Keele Street North York, Ontario, Canada M3J 1P3 and Fields Institute 222 College Street Toronto, Canada M5T 3J1

Fundamenta Mathematicae, Tome 196 (2007) no. 3, pp. 197-235

Résumé

If $G$ is a group then the abelian subgroup spectrum of $G$ is defined to be the set of all $\kappa$ such that there is a maximal abelian subgroup of $G$ of size $\kappa$. The cardinal invariant $A(G)$ is defined to be the least uncountable cardinal in the abelian subgroup spectrum of $G$. The value of $A(G)$ is examined for various groups $G$ which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the normal subgroup of permutations with finite support. It is shown that, if we use $G$ to denote this group, then $A(G) \leq \mathfrak a$. Moreover, it is consistent that $A(G) \neq \mathfrak a$. Related results are obtained for other quotients using Borel ideals.

Détail

DOI : 10.4064/fm196-3-1

Keywords: group abelian subgroup spectrum defined set kappa there maximal abelian subgroup size kappa cardinal invariant defined least uncountable cardinal abelian subgroup spectrum value examined various groups which quotients certain permutation groups integers important special which much paper devoted quotient full symmetric group normal subgroup permutations finite support shown denote group leq mathfrak moreover consistent neq mathfrak related results obtained other quotients using borel ideals