Groupes d'isom\'etries permutant doublement transitivement un ensemble de droites vectorielles

Publication date: 5 March 2009

Abstract: Let X be a non-empty finite set, E be a finite dimensional euclidean vector space and G a finite subgroup of O(E), the orthognal group of E. Suppose GG={U_i | i in X} is a finite set of linear lines in E and an orbit of G on which its operation is twice transitive. Then GG is an equiangular set of lines, which means that we can find a real number "c", and generators u_i of the lines U_i (i in X) such that forall i,j in X, ||u_i||=1, and if i is different from j then (u_i|u_j)=gve_{i,j}.c, with gve_{i,j} in {-1,+1} Let Gamma be the simple graph whose set of vertices is X, two of them, say i and j, being linked when gve_{i,j} = -1. In this article we first explore the relationship between double transitivity of G and geometric properties of Gamma. Then we construct several graphs associated with a twice transitive group G, in particular any of Paley's graphs is associated with a representation of G=PSL_2(q) on a set of q+1 equiangular lines in a vector space whose dimension is (q+1)/2.

Mathematics Subject Classification ID

Finite automorphism groups of algebraic, geometric, or combinatorial structures (20B25) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Graph representations (geometric and intersection representations, etc.) (05C62) General theory for finite permutation groups (20B05) Multiply transitive finite groups (20B20)

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