On finite \(p\)-groups with few nonabelian subgroups of order \(p^p\) and exponent \(p\).
The finite \(p\)-groups that are minimal nonabelian have been classified by \textit{L.~Rédei} [Comment. Math. Helv. 20, 225-264 (1947; Zbl 0035.01503)]. If \(G\) is a finite, nonabelian \(p\)-group, then its minimal nonabelian subgroups generate \(G\). The main result of this paper (Theorem~1.1) deals with a generalization of this fact. Let \(G\) be a finite, nonabelian group. Assume \(G\) is generated by elements of order at most \(p^k\), for some \(k\). Then \(G\) is generated by those minimal nonabelian subgroups, that are generated by elements of order at most \(p^{k+1}\). As an application, counting theorems for the number of certain subgroups of maximal class in a \(p\)-group are obtained.
scientific article; zbMATH DE number 5823087
scientific article