On a theorem of Blichfeldt

Let G be a permutation group on n<∞ objects. Let f(g) be the number of fixed points of g∈G, and let {f(g):1≠g∈G}={f1,…,fr}. In this expository note we give a character-free proof of a theorem of Blichfeldt which asserts that the order of G divides (n−f1)…(n−fr). We also discuss the sharpness of this bound.