Normal group algebras

Let \(\mathbb{F}G\) denote the group algebra of the group \(G\) over the field \(\mathbb{F}\) with \(char(\mathbb{F})\neq 2\). Given both a homomorphism \(\sigma:G\rightarrow \{\pm1\}\) and a group involution \(\ast: G\rightarrow G\), an oriented involution of \(\mathbb{F}G\) is defined by \(\alpha=\Sigma\alpha_{g}g \mapsto \alpha^\circledast=\Sigma\alpha_{g}\sigma(g)g^{\ast}\). In this paper, we determine the conditions under which the group algebra \(\mathbb{F}G\) is normal, that is, conditions under which \(\mathbb{F}G\) satisfies the \(\circledast\)-identity \(\alpha\alpha^\circledast=\alpha^\circledast\alpha\). We prove that \(\mathbb{F}G\) is normal if and only if the set of symmetric elements under \(\circledast\) is commutative.