# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 196 / 2007

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

#### Streszczenie

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.

#### Autorzy

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

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