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Subgroups of small Index in infinite Symmetric Groups
Author(s) -
Dixon John D.,
Neumann Peter M.,
Thomas Simon
Publication year - 1986
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/18.6.580
Subject(s) - von neumann architecture , queen (butterfly) , index (typography) , haven , library science , mathematics , combinatorics , computer science , pure mathematics , world wide web , hymenoptera , botany , biology
Throughout this paper Q will denote an infinite set, S:= Sym(Q) and G is a subgroup of S. If n: = |Q|, the cardinal of Q, then |.S| = 2". Working in ZFC, set theory with Axiom of Choice (AC), we shall be seeking the subgroups G with \S: G\ < 2". If A £ Q then S^ (respectively G{A}) denotes the setwise stabiliser of A in S (respectively in G); 5(A) and G(A) denote pointwise stabilisers; we identify S(A) with Sym(fi —A). A subset £ of Q such that |Z| = |Q — 1 | = |O| is known as a moiety ofQ. Suppose now that |Q| = n = Ko. If there is a finite subset A of Q such that 5(A) ^ G then certainly \S:G\ ^ Xo. Our theme is a rather strong converse: