Non-existence of abelian maximal subgroups in cyclic division algebras

TL;DR

This paper proves that cyclic division algebras do not have abelian maximal subgroups in their multiplicative groups, resolving a specific case of a longstanding conjecture and introducing a new criterion for maximal subgroups.

Contribution

It establishes the non-existence of abelian maximal subgroups in cyclic division algebras, advancing understanding of their subgroup structure and addressing a key conjecture.

Findings

No cyclic division algebra admits an abelian maximal subgroup.

Provides a new malnormality criterion for maximal subgroups.

Complements previous results on subgroup structures in 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.