On a theorem of Blichfeldt

TL;DR

This paper provides a clear, proof-free explanation of Blichfeldt's theorem relating permutation group order to fixed points, and examines the bound's optimality.

Contribution

It offers a character-free proof of Blichfeldt's theorem and discusses the bound's sharpness, enhancing understanding of permutation group properties.

Findings

The order of G divides the product of (n - fixed points counts).

The bound provided by Blichfeldt's theorem can be sharp.

The proof is accessible without character theory.

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.