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

TL;DR

This paper extends the understanding of the structure of unit groups in finite group algebras by analyzing the cases of $F(C_n imes Q_{12})$ and $F(C_n imes D_{12})$, generalizing previous specific studies.

Contribution

It establishes the structures of the unit groups of $F(C_n imes Q_{12})$ and $F(C_n imes D_{12})$ over finite fields, broadening prior results to more general cases.

Findings

Structures of unit groups for $F(C_n imes Q_{12})$ and $F(C_n imes D_{12})$ determined.

Generalization to finite fields of characteristic $p$ with $p^k$ elements.

Extends previous specific studies to more general group algebra cases.

Abstract

Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. The structures of the unit groups of the finite group algebras $FQ_{12}$ and $F(C_2 \times Q_{12})$ over a finite field $F$ have been studied in J. Gildea, F. Monaghan (2011), F. Monaghan (2012), G. Tang, Y. Gao (2011) and G. Tang, Y. Wei, Y. Li (2014) whereas the structures of the unit groups of the finite group algebras $FD_{12}$ and $F(C_2 \times D_{12})$ have been studied in J. Gildea, F. Monaghan (2011), N. Makhijani, R. K. Sharma, J. B. Srivastava (2016), F. Monaghan (2012), M. Sahai, S. F. Ansari and G. Tang, Y. Gao (2011). In this paper, we continue this study 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.