A Useful Property of the Finite Nonabelian Groups

TL;DR

This paper presents a new proof of Wedderburn's Little Theorem, demonstrating that finite division rings are commutative, by establishing a property of finite nonabelian groups involving maximally abelian subgroups.

Contribution

The paper introduces a novel property of finite nonabelian groups, showing that one of its maximally abelian subgroups is not an eigenheimer, and uses this to prove Wedderburn's Little Theorem.

Findings

Established a new property of finite nonabelian groups.

Provided a new proof of Wedderburn's Little Theorem.

Confirmed that finite division rings are commutative.

Abstract

In version v1 (under a different title) I was trying to give a new proof of Wedderburn's Little Theorem (WLT), stating that a finite dision ring is commutative, but I failed. So I had to withdraw the paper (version v2). Firstly I became aware of a new theorem. So in the mean time in version v3 under a new title (see top of this page) I proved a useful property of the finite nonabelian groups stating that one of its maximally abelian subgroups is not an eigenheimer. In the present version v4 I prove finally WLT in a new section 2 entitled: A New Proof of Wedderburn's Little Theorem: A Finite Division Ring is Commutative.