Group identities on symmetric units under oriented involutions in group algebras

TL;DR

This paper investigates conditions under which symmetric units in group algebras with oriented involutions satisfy group identities, linking these to polynomial identities and characterizing groups with nilpotent prime radicals.

Contribution

It proves that symmetric units satisfying a group identity imply the algebra satisfies a polynomial identity, affirming a conjecture of B. Hartley in this context.

Findings

Symmetric units satisfying a group identity imply the algebra satisfies a polynomial identity.

Characterization of groups with nilpotent prime radical where symmetric units satisfy a group identity.

Abstract

Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ over the infinite field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$, and let $\circledast:\mathbb{F}G\rightarrow \mathbb{F}G$ denote the involution defined by $\alpha=\Sigma\alpha_{g}g \mapsto \alpha^\circledast=\Sigma\alpha_{g}\sigma(g)g^{\ast}$, where $\sigma:G\rightarrow \{\pm1\}$ is a group homomorphism (called an orientation) and $\ast$ is an involution of the group $G$. In this paper we prove, under some assumptions, that if the $\circledast$-symmetric units of $\mathbb{F}G$ satisfies a group identity then $\mathbb{F}G$ satisfies a polynomial identity, i.e., we give an affirmative answer to a Conjecture of B. Hartley in this setting. Moreover, in the case when the prime radical $\eta(\mathbb{F}G)$ of $\mathbb{F}G$ is nilpotent we characterize the groups for which the symmetric units $\mathcal{U}^+(\mathbb{F}G)$ do satisfy a group identity.