Integral group rings with all central units trivial (Q1684745)

Let \(\mathbb QG\) be the rational group algebra of the finite group \(G\). There has been intensive research recently on central idempotents of group algebras of finite groups, such as in [\textit{A. Olivieri} et al., Commun. Algebra 32, No. 4, 1531-1550 (2004; Zbl 1081.20001)]; [\textit{G. K. Bakshi} and \textit{I. B. S. Passi}, Commun. Algebra 40, No. 4, 1413-1426 (2012; Zbl 1254.20003)] and [\textit{G. K. Bakshi} et al., J. Algebra Appl. 12, No. 3, Paper No. 1250168 (2013; Zbl 1348.20004)]. With an (extremely) strong Shoda pair \((H,K)\) of subgroups of \(G\) a central idempotent \(e(G,K,H)\) of the group algebra is associated. The main results are as follows. A complete irredundant set of extremely strong Shoda pairs is the set of pairs \((H,K)\) of subgroups of \(G\), where \(H\) is from a set of representatives of the set of all normal subgroups of the normal subgroup \(K\) under the equivalence relation defined by conjugacy of subgroups in \(G\) such that \(K/H\) is cyclic and is a maximal Abelian subgroup of \(N_G(H)/H\), \(K/\mathrm{core}(H)\) is an Abelian normal subgroup of maximal order in \(G/\mathrm{core}(H)\). Moreover, every primitive central idempotent of \(\mathbb QG\) is of the form \(e(G,K,H)\) for an extremely strong Shoda pair \((H,K)\) of \(G\) if and only if every complex irreducible character of \(G\) is induced from a linear character of a normal subgroup of \(G\). As an application, a complete set of primitive central idempotents and the Wedderburn decomposition of \(\mathbb QG\) are described for metabelian groups of order \(2sp^2\), \(p\) an odd prime and \(s\geq 1\), groups of order \(p^3\) and a non-metabelian group of order \(5^7\).