On the units of integral group ring of \(C_{n}\times C_{6}\)

Abstract

There are many kind of open problems with varying difficulty on units in a given integral group ring. In this note, we characterize the unit group of the integral group ring of \(C_{n}\times C_{6}\) where \(C_{n}=\langle a:a^{n}=1\rangle\) and \(C_{6}=\langle x:x^{6}=1\rangle\). We show that \(\mathcal{U}_{1}(\mathbb{Z}[C_{n}\times C_{6}])\) can be expressed in terms of its 4 subgroups. Furthermore, forms of units in these subgroups are described by the unit group \(\mathcal{U}_{1}(\mathbb{Z}C_{n})\). Notations mostly follow \cite{sehgal2002}.