Groups with a bounded number of conjugacy classes of non-normal subgroups
Let \(G\) be a group, and let \(\nu(G)\) be the number of conjugacy classes of non-normal subgroups of \(G\). The authors prove that, if \(G\) is a finite group with \(\nu(G)\neq 0\), then the centre \(Z(G)\) of \(G\) contains a cyclic subgroup \(C\) of prime power order such that the order of \(G/C\) is the product of at most \(\nu(G)+1\) primes. A similar statement can also be obtained for infinite groups with finitely many conjugacy classes of non-normal subgroups, but the subgroup \(C\) must be a Prüfer group. Moreover, the authors give a description of infinite groups with finitely many conjugacy classes of non-normal subgroups, generalizing a previous result by \textit{R. Brandl, S. Franciosi} and the reviewer [Proc. R. Ir. Acad., Sect. A 95, No. 1, 17-27 (1995; Zbl 0851.20019)].
scientific article; zbMATH DE number 1275253
scientific article