Ordered Permutation Groups

As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets.

A. M. W. Glass (Author)

9780521241908, Cambridge University Press

Paperback, published 7 January 1982

332 pages
22.8 x 15.2 x 1.9 cm, 0.499 kg

As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

Part I. Opening the innings: 1. Introduction
2. Doubly Transitive A
Part II. The structure theory: 3. Congruences and blocks
4. Primitive ordered permutation groups
5. The wreath product
Part III. Applications to ordered permutation groups: 6. Simple-permutation groups
7. Uniqueness of representation
8. Pointwise suprema and closed subgroups
Part IV. Applications to lattice-ordered groups: 10. Embedding theorums for lattice-ordered groups
11. Normal valued lattice-ordered groups
Part V. The author's perogative: 12. Algebraically closed lattice-ordered groups
13. The word problem for lattice-ordered groups.

Subject Areas: Mathematics [PB]