Products of Unipotent Elements in Certain Algebras

TL;DR

This paper characterizes when elements in group algebras over certain fields are products of unipotent elements, linking this to the derived subgroup and exploring properties of unipotent radicals, with applications to twisted group algebras.

Contribution

It provides a complete characterization of products of unipotent elements in group algebras over algebraically closed or positive characteristic fields, including bounds on the number of factors.

Findings

Elements in group algebras are products of unipotent elements iff they lie in the first derived subgroup.

Any such element can be expressed as a product of at most three unipotent elements.

The group algebra of a finite group over an infinite field cannot have a unipotent maximal subgroup.

Abstract

Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a product of unipotent elements if, and only if, it? lies in the first derived subgroup of the unit group of $FG$. In addition, it? is a product of at most three unipotent elements. Moreover, we explore some crucial properties satisfied by certain algebras like the connection between unipotent elements of index 2 and commutators as well as we investigate the unipotent radical of a group algebra by showing that the group algebra of a finite group over an infinite field cannot have a unipotent maximal subgroup. In particular, we apply these results to twisted group algebras.