The equations DkY = Xn in combinatorial species

AbstractThe category of combinatorial species was introduced by Joyal, and has been studied extensively by him and others. This category is equipped with a derivative operation (endofunctor). This allows one to differentiate species in a manner similar to differentiating power series. We solve, completely, the family of differential equations Dky = Xn for species. For this, we use results on the conjugacy classes of sharply k-transitive groups. We provide, also, a new proof of a sharpened version of Zassenhaus' theorem on sharply 2-transitive groups of type I