Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$ - Sivaranjani, Namatchivayam Umapathy; Nandakumar, Elumalai; Mittal, Gaurav; Sharma, Rajendra Kumar (Armenian Journal of Mathematics, 2024)
In this paper, we consider the general linear group $GL(2, 7)$ of $2 × 2$ invertible matrices over the finite field of order $7$ and compute the unit group of the semisimple group algebra $\mathbb{F}_qGL(2,7)$, where $\mathbb{F}_q$ is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of $\mathbb{F}_qGL(2,7)$, which is determined by first computing the Wedderburn decomposition of the group algebra $\mathbb{F}_q(PSL(3, 2)\rtimes C_2)$. Here $PSL(3,2)$ is the projective special linear group of degree 3 over a finite field of 2 elements.
Keywords: Unit Group; Group Algebra; General Linear Group; Wedderburn Decomposition