On maximal Subgroups of the multiplicative group of a division algebra

TL;DR

This paper investigates the conditions under which the multiplicative group of a division algebra contains maximal subgroups, revealing structural constraints related to the algebra's degree and properties of the center.

Contribution

It establishes necessary conditions for the existence of maximal subgroups in D*, linking algebra degree, divisibility, and the structure of division algebras over F.

Findings

If D* has no maximal subgroup, then deg(D) is not a power of 2.

F^{*2} is divisible.

Existence of noncyclic division algebras of degree p over F for each odd prime p dividing deg(D).

Abstract

The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.