On finite \(p\)-groups not satisfying the Hughes conjecture
type: Item
instance of: scholarly article
title: On finite \(p\)-groups not satisfying the Hughes conjecture
zbMATH Open document ID: 0835.20031
DOI: 10.1007/BF02104943
published in: Siberian Mathematical Journal
publication date: 1995-09-20T00:00:00Z
review text: The title refers to those finite \(p\)-groups for which the Hughes subgroup \(H_p(G):=\langle x\in G\mid x^p\neq 1\rangle\) is nontrivial and has index \(p^k>p\) in \(G\). Here, \(G\) stands for such a group. Let \(d\), \(m\) denote the minimal numbers of generators of \(G\), \(G/H_p(G)\) respectively and let \(c\), \(e\) be their respective nilpotency classes. It is shown (Theorem 2) that, if a multilinear identity of degree \(\leq(k- 1)(p-1)+1\) holds in the associated Lie ring of every finite group of exponent \(p\), then it also holds in the associated Lie ring of \(G\). It follows (Corollary 1) that \(c\leq k(d,p)\), where \(k(d,p)\) is the Kostrikin bound to the nilpotency class of a \(d\)-generator \((p-1)\)-Engel Lie ring of characteristic \(p\). It is also shown that, for each fixed \(p\), \(k(d,p)\) is bounded above by a linear function of \(d\). The following Theorem 4 (due to E. I. Zel'manov and the author) is deduced from Theorem 3 by a simple but ingenious argument: if the conjecture is true for a given \(p\), then the class \(e\) of \(G/H_p(G)\) is bounded for \(p\)-groups \(G\). With some further argument, it is deduced that \(G/H_p (G)\) must be abelian when \(p=5\).
reviewed by: G. E. Wall
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 794630
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Hughes subgroup
zbMATH Keywords: minimal numbers of generators
zbMATH Keywords: nilpotency classes
zbMATH Keywords: multilinear identities
zbMATH Keywords: associated Lie rings
zbMATH Keywords: nilpotency class
zbMATH Keywords: \((p-1)\)-Engel Lie rings
author: Evgenii I. Khukhro
MaRDI profile type: Publication
cites work: Q3825006
cites work: Q5719719
cites work: Q4711784
cites work: Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles
cites work: On a constrained variational problem and the space of horizontal paths
full work available at URL: https://doi.org/10.1007/bf02104943
OpenAlex ID: W2046838141
description: scientific article; zbMATH DE number 794630