Group identities on symmetric units under oriented involutions in group algebras

Abstract

Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ over the infinite field $\mathbb{F}$ with $char(\mathbb{F})\ne 2$, and let $⊛:\mathbb{F}G\to \mathbb{F}G$ denote the involution defined by $\alpha =\Sigma {\alpha }_{g}g\mapsto {\alpha }^{⊛}=\Sigma {\alpha }_g\sigma (g)g^{\ast }$, where $\sigma :G\to \{\pm 1\}$ 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 $⊛$-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.