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Non-existence of abelian maximal subgroups in cyclic division algebras

Huynh Viet Khanh

Abstract

We prove that no cyclic division algebra (in the sense of Dickson) admits an abelian maximal subgroup in its multiplicative group. This settles a special case of a long-standing conjecture of Akbari--Mahdavi-Hezavehi--Mahmudi and complements earlier results on locally nilpotent maximal subgroups and provides a new malnormality criterion for maximal subgroups.

Non-existence of abelian maximal subgroups in cyclic division algebras

Abstract

We prove that no cyclic division algebra (in the sense of Dickson) admits an abelian maximal subgroup in its multiplicative group. This settles a special case of a long-standing conjecture of Akbari--Mahdavi-Hezavehi--Mahmudi and complements earlier results on locally nilpotent maximal subgroups and provides a new malnormality criterion for maximal subgroups.

Paper Structure

This paper contains 4 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Main theorem
  4. Division algebras with cyclic maximal subfields

Key Result

Lemma 1

Let $D$ be a non-commutative division ring with centre $F = Z(D)$. If $M$ is an abelian maximal subgroup of $D^*$, then $K = M \cup \{0\}$ is a maximal subfield of $D$.

Theorems & Definitions (16)

  • Conjecture 1
  • Conjecture 2
  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 6 more