Browsing by Author "Thomas, S."
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Item Open Access The Cofinality Spectrum of the Infinite Symmetric Group(Cambridge University Press, 1997-09) Shelah, S.; Thomas, S.Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals A such that S can be expressed as the union of a chain of i proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. THEOREM. Suppose that V t GCH. Let C be a set of regular uncountable cardinals which satisfies the jollowing coalitions. (a) C contains a maximum element. (b) Iju is an inaccessible cardinal such that ui = sup(C n iu), then ,u E C. (c) I'li is a singular cardinal such that pi = sup(C n iu), then i + E C. Then there exists a ce..c. notion offorcing P such that VP t CF(S) = C. We shall also investigate the connections between the cofinality spectrum and pef theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.Item Open Access Infinite products of finite simple groups(American Mathematical Society, 1996) Saxl, J.; Shelah, S.; Thomas, S.We classify the sequences (Sn | n ∈ N) of finite simple nonabelian groups such that ∏n Sn has uncountable cofinality. © 1996 American Mathematical Society.Item Open Access Reducts of random hypergraphs(Elsevier BV, 1996-08-05) Thomas, S.For each k ⩾ 1, let Γk be the countable universal homogeneous k-hypergraph. In this paper, we shall classify the closed permutation groups G such that Aut(Γk) ⩽ G ⩽ Sym(Γk). In particular, we shall show that there exist only finitely many such groups G for each k ⩾ 1. We shall also show that each of the associated reducts of Γk is homogeneous with respect to a finite relational language.Item Open Access Some questions concerning the confinality of Sym (k)(Cambridge University Press, 1995-09) Sharp, J. D.; Thomas, S.