© The London Mathematical Society
Uncountable Cofinalities of Permutation Groups
Institut für Informatik, Universität Leipzig 04009 Leipzig, Germany droste{at}informatik.uni-leipzig.de
Fachbereich 6, Mathematik und Informatik, Universität Duisburg Essen 45117 Essen, Germany r.goebel{at}uni-essen.de
Received 17 December 2003. Revision received 13 July 2004.
A sufficient criterion is found for certain permutation groups G to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain (Hi)i
of proper subsets Hi such that Hi Hi 
Hi+1 and 
. This is a strong form of uncountable cofinality for G, where each Hi is a subgroup of G. This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym (Q) of all bounded permutations of the rationals Q which has uncountable cofinality but countable strong cofinality.
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  A. S. Kechris and C. Rosendal Turbulence, amalgamation, and generic automorphisms of homogeneous structures Proc. London Math. Soc., March 1, 2007; 94(2): 302 - 350. [Abstract] [Full Text] [PDF]
|
|