Speed’s Orbit Problem: Variations on Themes of Burnside

The paper offers a solution to T. P. Speed’s Orbit Problem, describing the orbit decomposition of direct powers of a transitive finite group action. The solution is given in terms of the Burnside algebra of the group and the right action of an incidence algebra of the subgroup lattice on a module of finitary functions. As a corollary, it is shown that almost all orbits in powers of a faithful action are regular. This gives one possible ‘GF(1)’ version of Burnside’s original theorem that each irreducible complex representation of a finite group appears in a tensor power of a fixed faithful representation.