Derivations of Non-Commutative Group Algebras

Abstract: In this article, we study the derivations of group algebras of some important groups, namely, dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($SD_{8n}$). First, we explicitly classify all inner derivations of a group algebra $\mathbb{F}G$ of a finite group $G$ over an arbitrary field $\mathbb{F}$. Then we classify all $\mathbb{F}$-derivations of the group algebras $\mathbb{F}D_{2n}$, $\mathbb{F}T_{4n}$ and $\mathbb{F}(SD_{8n})$ when $\mathbb{F}$ is a field of characteristic $0$ or an odd rational prime $p$ by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when $\mathbb{F}$ is an algebraic extension of a prime field.