Group algebras whose group of units is powerful

TL;DR

This paper investigates the structure of the group of units in group algebras over finite fields, showing it is only powerful when the underlying group is abelian, thus characterizing its algebraic properties.

Contribution

It proves that the group of units in the group algebra of a p-group over a finite field is powerful only if the group is abelian, providing a new structural insight.

Findings

Group of units is always a p-group

Group of units is powerful only if G is abelian

Characterization of the unit group's algebraic structure

Abstract

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.