Let G be a compact connected
Lie group which acts transitively and effectively on a sphere Sn−1. A manifold M is
said to have a sphere transitive structure if the structure group of the tangent bundle
of M can be reduced from 0(n) to G. The study of the existence of such structures is
a generalization of the well-known problem of the existence of almost complex
structures. We completely solve the question of existence of sphere transitive
structures on spheres. For our study of sphere transitive structures we need to know
some facts about the triality automorphism λ of Spin (8). We completely determine
the cohomology homomorphism induced by λ on the cohomology of the classifying
space of Spin (8).