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A generalization of a result of Hegedüs

Sara C. Debón

Abstract

A famous result of P. Hegedüs describes the structure of finite rational $\{2,5\}$-groups. In this paper, we show how Hegedüs' 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 $p\geq 5$.

A generalization of a result of Hegedüs

Abstract

A famous result of P. Hegedüs describes the structure of finite rational -groups. In this paper, we show how Hegedüs' proof can be used to obtain a similar result for the more general situation of a finite rational -group acting faithfully and with the eigenvector property on a f.d -vector space with .
Paper Structure (3 sections, 4 theorems, 14 equations)

This paper contains 3 sections, 4 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Proof of Theorem A

Key Result

Theorem 1.1

(Theorem 1.2 of Hegedus2005) If $H$ is a finite rational $\{2, 5\}$-group, then $P\in \operatorname{Syl}_5(H)$ is normal and $P\cong C_5^{2n}$ for some $n\in \mathbb{N}$. Moreover, $H/O_2(H)\cong \mathbb{M}\wr K$ for $K$ a $2$-subgroup of $\Sigma_n$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 1 more