A generalization of Schmidt's theorem on groups with all subgroups nilpotent.
The author studies finite \(\mathcal S^n\)-groups, which are defined as the groups having exactly \(n\) non-nilpotent subgroups. This generalises the minimal non-nilpotent groups or Schmidt groups, which turn out to be the \(\mathcal S^1\)-groups. The main results of the paper are that finite \(\mathcal S^n\)-groups with \(n<22\) are soluble and that the alternating group of degree~\(5\) is an \(\mathcal S^{22}\)-group, and that the only finite non-Abelian semisimple (that is, with no non-trivial normal Abelian subgroups) \(\mathcal S^n\)-groups with \(n\leq 65\) are the alternating and the symmetric group of degree~\(5\) and the simple group of order~\(168\).
scientific article; zbMATH DE number 6109275