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Subgroups of Infinite Symmetric Groups
Author(s) -
Macpherson H. D.,
Neumann Peter M.
Publication year - 1990
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-42.1.64
Subject(s) - queen (butterfly) , von neumann architecture , citation , section (typography) , mile , combinatorics , library science , mathematics , computer science , physics , pure mathematics , hymenoptera , botany , astronomy , biology , operating system
This paper and its sequel [17] deal with a range of questions about the subgroup structure of infinite symmetric groups. Our concern is with such questions as the following. How can an infinite symmetric group be expressed as the union of a chain of proper subgroups? What are the subgroups that supplement the normal subgroups of an infinite symmetric group? What are the maximal proper subgroups? Is every proper subgroup contained in a maximal proper subgroup? Although our observations about maximal subgroups are inconclusive we shall show that there are unexpected relationships between these and other questions about infinite symmetric groups. Throughout the paper Q will denote an infinite set of cardinality K and S, S(K) or Sym (Q) will denote the symmetric group on Q. In §3 we consider chains (ordered by inclusion of course) of proper subgroups of S. Our main result is the following.