Book contents
- Frontmatter
- Contents
- Introduction
- List of Talks
- Participants
- Some Recent Combinatorial Applications of Borsuk-Type Theorems
- On Extremal Finite Sets in the Sphere and Other Metric Spaces
- Metric and Geometric Properties of Sets of Permutations
- Infinite Geometric Groups and Sets
- Intersection and Containment Problems Without Size Restrictions
- Distance-Transitive Graphs of Valency k, 8 ≤ k ≤ 13
- Latin Square Determinants
- A Computer Search for a Projective Plane of Order 10
- Matroids, Algebraic and Non Algebraic
- Algebraic Properties of a General Convolution
- Quasi Groups, Association Schemes, and Laplace Operators on Almost Periodic Functions
- Geometric Methods in Group Theory
- Problem Section
Infinite Geometric Groups and Sets
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Introduction
- List of Talks
- Participants
- Some Recent Combinatorial Applications of Borsuk-Type Theorems
- On Extremal Finite Sets in the Sphere and Other Metric Spaces
- Metric and Geometric Properties of Sets of Permutations
- Infinite Geometric Groups and Sets
- Intersection and Containment Problems Without Size Restrictions
- Distance-Transitive Graphs of Valency k, 8 ≤ k ≤ 13
- Latin Square Determinants
- A Computer Search for a Projective Plane of Order 10
- Matroids, Algebraic and Non Algebraic
- Algebraic Properties of a General Convolution
- Quasi Groups, Association Schemes, and Laplace Operators on Almost Periodic Functions
- Geometric Methods in Group Theory
- Problem Section
Summary
ABSTRACT
We investigate geometric groups and sets of permutations of an infinite set. (These are a generalisation of sharply t–transitive groups and sets). We prove non–existence of groups, and give constructions of sets, for certain parameters. This work was done while the authors were visiting the Ohio State University, to whom we express our gratitude.
INTRODUCTION
It is known that sharply t–transitive groups of permutations of an infinite set exist only for t ≤ 3 (Tits (1952)), while sharply t–transitive sets exist for al l t (Barlotti & Strambach 1984).
Geometric groups and sets of permutations have been proposed as a natural generalisation of sharply t–transitive groups and sets (Cameron & Deza (1979)). Our purpose i s to investigate such objects on infinite sets. Not surprisingly, we give nonexistence results for groups, and constructions for sets.
Let L = {ℓ0, ℓt …, ℓs–1) be a finit e set of natural numbers, with ℓ0 < … < ℓs–1. The permutation group G on the set X is a geometric group of type L if there exist points x1, …, xs ϵ X such that
(i) the stabiliser of x1 …, xs is the identity;
(ii) for i < s, the stabiliser of x1 …, xi fixes ℓi points and acts transitively on its non–fixed points.
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- Chapter
- Information
- Algebraic, Extremal and Metric Combinatorics 1986 , pp. 54 - 61Publisher: Cambridge University PressPrint publication year: 1988