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

Volume 196 / 2007

Saharon Shelah, Juris Steprāns Fundamenta Mathematicae 196 (2007), 197-235 MSC: 03E17, 03E35, 03E40, 03E50, 20B07, 20B30, 20B35. DOI: 10.4064/fm196-3-1


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.


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

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