Group algebra whose unit group is locally nilpotent

TL;DR

This paper classifies groups and fields where the group algebra's unit group is locally nilpotent or Engel, and where the algebra has finitely many nilpotent elements, advancing understanding of algebraic structures with specific nilpotency properties.

Contribution

It provides a complete classification of groups and fields for which the unit group of the group algebra exhibits local nilpotency or Engel properties, and finitely many nilpotent elements.

Findings

Identifies all groups and fields with locally nilpotent unit groups

Characterizes cases where the algebra has finitely many nilpotent elements

Determines when the unit group is an Engel group

Abstract

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.