On a theorem of Blichfeldt

Abstract

Let $G$ be a permutation group on $n<\infty$ objects. Let $f(g)$ be the number of fixed points of $g\in G$, and let $\{f(g):1\ne g\in G\}=\{f_1,\ldots,f_r\}$. In this expository note we give a character-free proof of a theorem of Blichfeldt which asserts that the order of $G$ divides $(n-f_1)\ldots(n-f_r)$. We also discuss the sharpness of this bound.