矢量群变换二重传递群
Groupes d'isométries permutant doublement transitivement un ensemble de droites vectorielles
摘要：
设X是非空有限集，E是有限维欧氏向量空间，G是E的正交群O（E）的有限子群.假设GG={U_i| i in X}是E中线性直线的有限集，是G的一个轨道，其上的运算是二次传递的。则GG是一组等角直线，这意味着我们可以找到一个实数“c”，以及直线U_i（i in X）的生成元u_i，使得对于X中的所有i，j，||u_i|| =1，如果i不同于j，则（u_i| u_j）= \gve_{i,j} .c，其中\gve_{i,j}在{-1，+1\}中设Gamma是顶点集为X的单图，当\gve_{i,j} =-1时，其中两个顶点（比如i和j）是连通的.本文首先探讨了G的双传递性与Gamma的几何性质之间的关系。然后我们构造了几个与二次传递群G相关联的图，特别是在维数为（q+1）/2的向量空间中，任何Paley图都与G=PSL_2（q）在q+1条等角线上的表示相关联.
英文摘要：
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.
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