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

Saharon Shelah, and Juris Steprāns. "Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers." Fundamenta Mathematicae 196.3 (2007): 197-235. <http://eudml.org/doc/282620>.

@article{SaharonShelah2007,
abstract = {If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. 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) ≤ 𝔞. Moreover, it is consistent that A(G) ≠ 𝔞. Related results are obtained for other quotients using Borel ideals.},
author = {Saharon Shelah, Juris Steprāns},
journal = {Fundamenta Mathematicae},
keywords = {quotients of groups; consistency; abelian subgroup spectrum; cardinal invariant; Borel ideals},
language = {eng},
number = {3},
pages = {197-235},
title = {Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers},
url = {http://eudml.org/doc/282620},
volume = {196},
year = {2007},
}

TY - JOUR
AU - Saharon Shelah
AU - Juris Steprāns
TI - Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers
JO - Fundamenta Mathematicae
PY - 2007
VL - 196
IS - 3
SP - 197
EP - 235
AB - If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. 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) ≤ 𝔞. Moreover, it is consistent that A(G) ≠ 𝔞. Related results are obtained for other quotients using Borel ideals.
LA - eng
KW - quotients of groups; consistency; abelian subgroup spectrum; cardinal invariant; Borel ideals
UR - http://eudml.org/doc/282620
ER -