# Normal group algebras

## TL;DR

This paper characterizes when a group algebra over a field with characteristic not 2 is normal under an oriented involution, linking normality to the commutativity of symmetric elements.

## Contribution

It provides necessary and sufficient conditions for the normality of group algebras with respect to a specific involution involving a homomorphism and a group involution.

## Key findings

- Normality of the algebra is equivalent to the commutativity of symmetric elements.
- The paper establishes a clear criterion for normality based on symmetric elements.
- Conditions are given for when the algebra satisfies the ircledast-identity.

## Abstract

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.