Groups of central units of rank 1 of integral group rings of Frobenius metacyclic groups (Q2033368)

Let \(\mathbb ZG\) be the integral group ring of a group \(G\) and let \(F_{mn}=\left\langle b\right\rangle_m\leftthreetimes\left\langle a\right\rangle_n\) be a Frobenius metacyclic group of order \(mn\) with a kernel \(\left\langle b\right\rangle\) of order \(m\) and a complement \(\left\langle a\right\rangle\) of order \(n\). The generating elements of the group of central units of integral group rings of Frobenius metacyclic groups of orders 10 and 55 were found in [\textit{E. O. Shumakova}, Sib. Èlektron. Mat. Izv. 18, 622--639 (2021; Zbl 07360172)] and of order 78 in [\textit{E.O. Shumakova}, ``A description of the group of central units in integral group rings of Frobenius groups'', Innov. Sci. 7, 60--65 (2016)] In this paper, the author gives a description of the central units of the integral group rings \(\mathbb ZF_{13,3}\) (Theorem 1) and \(\mathbb ZF_{13,12}\) (Theorem 2). In view of the above results, the author obtains a description of the group of the central units of a rank 1 of integral group ring \(\mathbb ZF_{mn}\), when \(m\) is a prime.