Finite p-groups that are close to groups of prime period
The existence of counterexamples to conjectures about the generalized Hughes subgroup suggested by \textit{N. Blackburn} and \textit{A. Espuelas} [Proc. Am. Math. Soc. 92, 478-484 (1984; Zbl 0556.20016)] and \textit{A. Espuelas} [Arch. Math. 43, 388-390 (1984; Zbl 0534.20014)] is proved. This follows from the existence of counterexamples to the Hughes conjecture [\textit{G. E. Wall}, Lect. Notes Math. 372, 667-690 (1974; Zbl 0286.20050)] and the following theorem of the author: Suppose that for every \(k=1,2,...,r\) there is a multilinear identity of degree \(1+k(p-1)\) which holds in the Lie ring of a free group of prime exponent p and not follows from identities of smaller degree. Then there exists a finite nilpotent group P of class \(1+r(p-1)\) such that the subgroup \(<x^ p| x\in P>\) has order p, the Frattini subgroup \(\Phi\) (P) coincides with \(\Omega_ 1(P)=<x\in P|\) \(x^ p=1>\) and \(| P:\Phi (P)| \geq p^{1+r(p-1)}\).
scientific article; zbMATH DE number 3993838