Ordered Groups and Infinite Permutation Groups: 354 (Mathematics and Its Applications, 354) - Hardcover

Hardcover

ISBN 10: 0792338537 ISBN 13: 9780792338536

Publisher: Springer, 1995

Synopsis

The subjects of ordered groups and of infinite permutation groups have long en­ joyed a symbiotic relationship. Although the two subjects come from very different sources, they have in certain ways come together, and each has derived considerable benefit from the other. My own personal contact with this interaction began in 1961. I had done Ph. D. work on sequence convergence in totally ordered groups under the direction of Paul Conrad. In the process, I had encountered "pseudo-convergent" sequences in an ordered group G, which are like Cauchy sequences, except that the differences be­ tween terms of large index approach not 0 but a convex subgroup G of G. If G is normal, then such sequences are conveniently described as Cauchy sequences in the quotient ordered group GIG. If G is not normal, of course GIG has no group structure, though it is still a totally ordered set. The best that can be said is that the elements of G permute GIG in an order-preserving fashion. In independent investigations around that time, both P. Conrad and P. Cohn had showed that a group admits a total right ordering if and only if the group is a group of automor­ phisms of a totally ordered set. (In a right ordered group, the order is required to be preserved by all right translations, unlike a (two-sided) ordered group, where both right and left translations must preserve the order.

About this title

Synopsis

The separate areas of Ordered Groups and Infinite Permutation Groups began to converge in significant ways about thirty years ago. Since then, the connection has steadily grown so that now permutation groups are essential to many who work in ordered groups. Ordered groups are of some interest to most of those who work in infinite permutation groups, and there are a number of mathematicians whose main work is exactly in ordered permutation groups, the combination of the two. This book represents the happy confluence of the two subjects, running the spectrum from purely infinite permutation groups through ordered permutation groups to purely ordered groups.Experts in various aspects of these subjects have each contributed a chapter. The articles are surveys of recent and past work in the area and they include extensive bibliographies. The topics include lattice-ordered groups, ordered permutation groups, Jordan groups, reconstruction problems, groups with few orbits, the separation theorem, and automorphisms of symmetric groups. This book is an essential reference for anyone working in ordered groups or infinite permutation groups.

Edition details

• Publisher: Springer
• Publication date: 1995
• Language: English
• ISBN 10: 0792338537
• ISBN 13: 9780792338536
• Binding: Hardcover
• Number of pages: 256
• Editor: Holland W.C.