Groups with many subnormal subgroups (Q1399174)

According to a classical result of Roseblade there is a function \(\rho\) such that all groups with all subgroups subnormal of defect at most \(n\) are nilpotent of class at most \(\rho(n)\). The groups considered here have the property that all subgroups containing a fixed finite subgroup \(F\) are subnormal of defect at most \(n\). Main results: If \(G\) is locally nilpotent, there is a function \(\beta\) such that \(G_{\beta(n)}\) is finite (and so \(G\) is nilpotent of a class depending on \(n\) and \(o(F)\)). -- If \(G\) is a torsion group and only finitely many primes \(p\) are orders of group elements, then \(G_{\beta(n)}\) is finite (Theorems 0.2 and 0.3). The finiteness of the set of primes is indispensable; this is shown by a counterexample.