On the Unit Conjecture for Supersoluble Group Rings, I

TL;DR

This paper develops new structure theorems for supersoluble group rings to analyze the unit conjecture, demonstrating the absence of certain units in specific group algebras and introducing a framework for future research.

Contribution

It introduces structure theorems for supersoluble group rings and applies them to show non-existence of certain units in group algebras of the Passman fours group.

Findings

No non-trivial units of length ≤ 3 in KG

Promislow set cannot be the support of a unit in KG

Preliminary analysis of higher-length units using consistent chains

Abstract

We introduce structure theorems for the study of the unit conjecture for supersoluble group rings and apply our results to the (Passman) fours group G. We show that over any field K, the group algebra KG has no non-trivial units of length at most 3, and find that the Promislow set can never be the support of a unit in KG. We conclude our work with an introduction to the theory of "consistent chains" toward a preliminary analysis of units of higher length in KG.