On exceptional \(pq\)-groups (Q1934001)

Let \(F/k\) be a normal extension of number fields with Galois group \(G\). If in the Brauer-Kuroda relation of the Dedekind zeta functions of intermediate fields of \(F/k\) the zeta function of \(F\) does not appear (this only depends on \(G\)), then the group \(G\) is said to be exceptional (see \textit{J. Browkin} et al. [Bull. Pol. Acad. Sci., Math. 59, No. 3, 207--214 (2011; Zbl 1244.20013)]). In this article, the authors investigate the exceptional groups whose orders are products of powers of two primes \(p\) and \(q\) such that \(G\) has unique subgroups of orders \(p\) and \(q\). More precisely, let \(G\) be as above, let \(N_d(G)\) denote the number of cyclic subgroups of \(G\) with order \(d\), and set \(N(G) = (N_p(G),N_q(G), N_{pq}(G))\). Thm.~4.1 shows that if \(N_p(G) = N_q(G) = 1\), then the Sylow subgroups of \(G\) are cyclic or generalized quaternion. This is used in Thm.~4.5. to classify all such groups \(G\) with \(N(G) = (1,1,1)\).