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

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.