Group identities on symmetric units under oriented involutions in group algebras

Abstract

AbstractLet $$\mathbb {F}G$$ F G denote the group algebra of a locally finite group G over the infinite field $$\mathbb {F}$$ F with $$\mathop {\textrm{char}}\nolimits (\mathbb {F})\ne 2$$ char ( F ) ≠ 2 , and let $$\circledast :\mathbb {F}G\rightarrow \mathbb {F}G$$ ⊛ : F G → F G denote the involution defined by $$\alpha =\Sigma \alpha _{g}g \mapsto \alpha ^\circledast =\Sigma \alpha _{g}\sigma (g)g^{*}$$ α = Σ α g g ↦ α ⊛ = Σ α g σ ( g ) g ∗ , where $$\sigma :G\rightarrow \{\pm 1\}$$ σ : G → { ± 1 } is a group homomorphism (called an orientation) and $$*$$ ∗ is an involution of the group G. In this paper we prove, under some assumptions, that if the $$\circledast $$ ⊛ -symmetric units of $$\mathbb {F}G$$ F G satisfies a group identity then $$\mathbb {F}G$$ 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 $$\eta (\mathbb {F}G)$$ η ( F G ) of $$\mathbb {F}G$$ F G is nilpotent we characterize the groups for which the symmetric units $$\mathcal {U}^+(\mathbb {F}G)$$ U + ( F G ) do satisfy a group identity.