The Structure of the Unit Group of $\mathbb{F}_{3^{k}}(C_{3}\times D_{2n})$ and Finite Group Algebras of Groups of Order $42$

Year 2024, Early Access, 1 - 19

Sandeep Malik Rajendra Kumar Sharma

https://doi.org/10.24330/ieja.1571446

Abstract

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p>0$ with $|\mathbb{F}_{q}|=q=p^{k}$ and $\mathcal{U}(\mathbb{F}_{q}G)$ be the unit group of the group algebra $\mathbb{F}_{q}G$ for some group $G$. There are $6$ groups of order $42$ up to isomorphism. In this paper, we provide a characterization of $\mathcal{U}(\mathbb{F}_{3^{k}}(C_{3}\times D_{2n}))$ and establish the structures of the unit groups of some finite group algebras of groups of order $42$.

Keywords

Group algebra, Jacobson radical, Wedderburn decomposition, unit group

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