On the unit groups of the integral group rings of metacyclic groups (Q1104391)

One method to tackle the so-called isomorphism problem, namely does \({\mathbb{Z}}G\cong {\mathbb{Z}}H\) imply \(G\cong H\), consists in constructing a normal complement C of G in the group of units of \({\mathbb{Z}}G\) which map to 1 in \({\mathbb{Z}}\). The best result achieved so far is the one [in J. Algebra 73, 167-185 (1981; Zbl 0484.16004)], due to \textit{G. H. Cliff}, \textit{S. K. Seghal} and \textit{A. R. Weiss}, which states that C indeed exists for certain metabelian groups G. On the other hand, \textit{K. W. Roggenkamp} and \textit{L. L. Scott} [J. Pure Appl. Algebra 27, 299-314 (1983; Zbl 0509.16006)] produced metabelian groups that do not have a normal complement C. The present paper pursues these investigations and proves that for split metacyclic groups of type \(p\rtimes m\), where m is even, \(m\dag 12\), and p is a prime, a normal complement C almost never (with respect to p) exists.