A generalization of a result of Heged\"us

TL;DR

This paper extends Heged"us' classical result on finite rational groups to a broader context involving finite rational 2-groups acting on finite-dimensional vector spaces over finite fields, revealing new structural insights.

Contribution

It generalizes Heged"us' theorem to include finite rational 2-groups acting faithfully with the eigenvector property on finite fields, broadening the scope of the original result.

Findings

Heged"us' proof technique applies to more general group actions.

Finite rational 2-groups exhibit similar structural properties in broader settings.

The eigenvector property plays a key role in the generalization.

Abstract

A famous result of P. Heged\"us describes the structure of finite rational $\{2,5\}$-groups. In this paper, we show how Heged\"us' proof can be used to obtain a similar result for the more general situation of a finite rational $2$-group acting faithfully and with the eigenvector property on a f.d $\mathbb{F}_p$-vector space with $p\geq 5$.