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A Block-theoretic Proof of Burnside's Normal $p$-complement Theorem

Christopher Herbig

Abstract

In [3, Theorem 6.7B], the authors use the Main Theorems of Brauer to give a proof of Burnside's Normal $p$-complement Theorem. Unfortunately, the proof contains an error. We take this opportunity to give a proof along similar lines, circumventing the error by means of a well-known result on traces of totally positive cyclotomic integers.

A Block-theoretic Proof of Burnside's Normal $p$-complement Theorem

Abstract

In [3, Theorem 6.7B], the authors use the Main Theorems of Brauer to give a proof of Burnside's Normal -complement Theorem. Unfortunately, the proof contains an error. We take this opportunity to give a proof along similar lines, circumventing the error by means of a well-known result on traces of totally positive cyclotomic integers.

Paper Structure

This paper contains 3 sections, 7 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas
  3. The Proof of the Theorem

Key Result

Theorem 1.1

(Burnside) Let $G$ be a finite group. If $C_G(P) = N_G(P)$ for $P \in \mathop{\mathrm{Syl}}\nolimits_{p}(G)$, then $G$ has a normal $p$-complement.

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 2 more