Lie nilpotency indices of symmetric elements under oriented involutions in group algebras

TL;DR

This paper investigates bounds on the Lie nilpotency index of symmetric elements and the nilpotency class of symmetric units in group algebras with oriented involutions, providing new theoretical insights.

Contribution

It introduces bounds on Lie nilpotency indices and nilpotency classes for symmetric elements and units under oriented involutions in group algebras, a novel theoretical contribution.

Findings

Established bounds on the Lie nilpotency index of symmetric elements

Derived bounds on the nilpotency class of symmetric units

Provided theoretical framework for understanding symmetric elements in group algebras

Abstract

Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.