Finite p-groups that are close to groups of prime period
type: Item
instance of: scholarly article
title: Finite p-groups that are close to groups of prime period
zbMATH Open document ID: 0614.20010
DOI: 10.1007/BF01978887
published in: Algebra and Logic
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187347
review text: 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 \(| P:\Phi (P)| \geq p^{1+r(p-1)}\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 3993838
zbMATH Keywords: generalized Hughes subgroup
zbMATH Keywords: Hughes conjecture
zbMATH Keywords: multilinear identity
zbMATH Keywords: Lie ring
zbMATH Keywords: free group of prime exponent
zbMATH Keywords: Frattini subgroup
author: Evgenii I. Khukhro
reviewed by: V. D. Mazurov
MaRDI profile type: Publication
cites work: The Power Structure of Metabelian p-Groups
cites work: On the Hughes problem for exponent \(p^ 2\)
cites work: The Hughes problem for exponent nine
cites work: The \(H_p\)-problem and the structure of \(H_p\)-groups
cites work: On the Lie ring of a group of prime exponent II
cites work: On Finite Groups With ‘Hidden’ Primes
cites work: Secretive prime-power groups of large rank
cites work: Démonstration d'une hypothèse de M. Artin
description: scientific article; zbMATH DE number 3993838