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

1. (i) the stabiliser of x1 …, xs is the identity;

2. (ii) for i < s, the stabiliser of x1 …, xi fixes ℓi points and acts transitively on its non–fixed points.

INTRODUCTION

After the definition of matroids in the 30's a lot of work has been done on vector representations of matroids. Much less work has been devoted to algebraic representations of matroids in the proper sense: after the pioneer work of S. MacLane fifty years ago comes the work of A.W. Ingleton et al. fifteen years ago. Not more than five years ago I decided to try to solve some of the open problems in this neglected field. Recently more people have become interested in it. I intend to give a survey of contributions I know about.

First I recall some definitions. For a better introduction I recommend Chapter 11 of the book of Welsh (1976).

Let F be a fixed field and K an extension of F. Elements e1,…,en in K are algebraically dependent over F when there is a non-zero polynomial p(X1,…,Xn) ϵ F[X] such that p(e1…,en) = 0. If E is a finite subset of K then the algebraically independent subsets of E over F give the independent sets of a matroid M(E). Such a matroid is called algebraic. The rank r(A) of a subset A ⊆ E in this matroid is the transcendence degree tr.d.F F(A) of the field F(A) over F.

INTRODUCTION

There is a well–known and intimate connection between the character theory and the ordinary representation theory of a finite group: characters are traces of matrix representations, and determine the representations up to equivalence. For a finite non–empty quasigroup Q, the connections are much more obscure. The character theory is that of the association scheme (Q×Q; C1,…,Cs) determined by the permutation representation of the combinatorial multiplication group on the quasigroup [1, pp. 181·2] [3] [6]. The representation theory is that of free groups determined by the quasigroup – the universal multiplication group and point stabilizers within it [6]. Representations are classified by almost periodic functions on these free groups. The character theory furnishes generalized Laplace operators Δ1,…,Δs acting on the almost periodic functions.

ACKNOWLEDGEMENT

This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grants A9373, 0011 and by the Fonds pour la Formation de Chercheurs et l'Aide á la Recherche under Grant EQ2369.

INTRODUCTION

A finite projective plane of order n is a collection of n2+n+1 lines and n2+n+1 points such that

1. (1.1) every line contains n+1 points,

2. (1.2) every point is on n+1 lines,

3. (1.3) any two distinct lines intersect at exactly one point, and

4. (1.4) any two distinct points lie on exactly one line.

For example, a projective plane of order 2 is shown in Fig. 1. It has 7 points and 7 lines. The points are numbered from 1 to 7. The 7 lines are L1 = {1,2,4}, L2= {2,3,5}, L3 = {3,4,6}, L4= {4,5,7}, L5 {1,5,6}, L6 {2,6,7} and L7 = {1,3,7}. In Fig. 1 all the lines except L6 are drawn as straight lines. One can easily show that this projective plane of order 2 is unique up to the relabelling of points and lines.

Another way to represent a projective plane is to use an incidence matrix A of size n2 +n+ 1 by n2+n+1.