Through a combination of linear algebra, geometry, and algebraic structures, one can prove that certain families of geometric designs have groups of automorphisms that are doubly transitive. These geometric designs can be defined as edge colorings of complete graphs. Automorphisms of these geometric designs are permutations of the vertices of the graphs, which are also permutations of the colors of the edges such that given two edges of the same color, their images are also the same color. All cases considered are on the vector space Fn for a field F and the group of automorphisms is a subgroup of the affine general linear group AGL (F,n). Three families of these designs have been explored. The first family is derived from an absolute value, the second from the addition of vectors, and the third from an inner product.
Persons, Michelle, "An Exploration of Doubly Transitive Designs in Affine Space" (1995). Honors Theses. 530.