Download e-book for iPad: Groups St Andrews 2005: Volume 1 by C. M. Campbell, M. R. Quick, E. F. Robertson, G. C. Smith

By C. M. Campbell, M. R. Quick, E. F. Robertson, G. C. Smith

ISBN-10: 0521694698

ISBN-13: 9780521694698

'Groups St Andrews 2005' was once held within the collage of St Andrews in August 2005 and this primary quantity of a two-volume booklet includes chosen papers from the foreign convention. 4 major lecture classes got on the convention, and articles in response to their lectures shape a considerable a part of the complaints. This quantity includes the contributions through Peter Cameron (Queen Mary, London) and Rostislav Grogorchuk (Texas A&M, USA). except the most audio system, refereed survey and examine articles have been contributed through different convention individuals. prepared in alphabetical order, those articles conceal a large spectrum of contemporary crew thought. The commonplace complaints of teams St Andrews meetings have supplied snapshots of the kingdom of analysis in workforce thought during the previous 25 years. previous volumes have had a tremendous impression at the improvement of workforce conception and it's expected that this quantity should be both very important.

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Richman, Maximal subgroups of infinite symmetric groups, Canad. Math. Bull. 10 (1967), 375–381. [57] J. H. Schmerl, Countable homogeneous partially ordered sets, Algebra Universalis 9 (1979), 317–321. [58] L. L. Scott, Representations in characteristic p, Proc. Symp. Pure Math. 37 (1980), 319–331. [59] S. Shelah and S. Thomas, The cofinality spectrum of the infinite symmetric group, J. Symbolic Logic 62 (1997), 902–916. [60] S. R. Thomas, Reducts of the random graph, J. Symbolic Logic 56 (1991), 176–181.

Note also that Macpherson and Neumann proved that a chain of proper subgroups with union Sym(Ω) must have length greater than |Ω|, so in a sense the result of Baumgartner et al. ) If H is a maximal subgroup of Sym(Ω), then H, g = Sym(Ω) for all g ∈ / H. Galvin [30] proved the remarkable result that if H is any subgroup such that H, B = Sym(Ω) for some set B with |B| ≤ |Ω|, then there is some element g such that H, g = Sym(Ω). Moreover, the order of g can be chosen to be any preassigned even number greater than 2.

Peter Neumann raised the question: is every proper subgroup H of Sym(Ω) contained in a maximal subgroup? Macpherson and Praeger showed that this is the case if H is not highly transitive. However, Baumgartner et al. [4] showed that it is consistent with the ZFC axioms for set theory that there exists a subgroup T of Sym(Ω), where |Ω| = κ, for which the subgroups between T and Sym(Ω) form a well-ordered chain of order type κ+ . (This result is proved assuming GCH. Of course, all the intermediate groups are highly transitive.

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