Units in $F(C_n \\times Q_{12})$ and $F(C_n \\times D_{12})$

Authors : Sheere Farhat ANSARI, Meena SAHAI

Pages : 182-196

Doi:10.24330/ieja.1299278

View : 14 | Download : 26

Publication Date : 2023-07-10

Article Type : Research Paper

Abstract :Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of $2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, n\\geq 3$ and the generalized quaternion groups $Q_{4n}, n\\geq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $Finsert ignore into journalissuearticles values(C_2 \\times Q_{12});$ and $Finsert ignore into journalissuearticles values(C_2 \\times D_{12});$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $Finsert ignore into journalissuearticles values(C_n \\times Q_{12});$ and $Finsert ignore into journalissuearticles values(C_n \\times D_{12});$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.
Keywords : Group algebra, unit group, dihedral group, cyclic group, quaternion group