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

Year 2023, , 182 - 196, 10.07.2023

Sheere Farhat Ansarı Meena Sahaı

https://doi.org/10.24330/ieja.1299278

Cited By: 1

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 $FQ_{12}$, $FD_{12}$, $F(C_2 \times Q_{12})$ and $F(C_2 \times D_{12})$ 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 $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.

Keywords

Group algebra, unit group, dihedral group, cyclic group, quaternion group

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