On counterexamples to the Hughes conjecture.
type: Item
instance of: scholarly article
title: On counterexamples to the Hughes conjecture.
zbMATH Open document ID: 1187.20010
author: George Havas
author: Michael Vaughan-Lee
published in: Journal of Algebra
publication date: 2009-10-02T00:00:00Z
review text: Let \(G\) be a finite group and \(p\) a prime. The \(p\)-Hughes subgroup \(H_p(G)\) is generated by all the elements of \(G\) which do not have order \(p\). The Hughes problem (1957) asks whether \(|G:H_p(G)|=p\) whenever \(1\neq H_p(G)\neq G\). It was solved in the affirmative for \(G\) being not a \(p\)-group [\textit{D. R. Hughes} and \textit{J. G. Thompson}, Pac. J. Math. 9, 1097-1101 (1959; Zbl 0098.25201)], as well as for some classes of \(p\)-groups. But \textit{G. E. Wall} [Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1965; Zbl 0189.31701)] constructed a 3-generator counterexample (``anti-Hughes \(p\)-group'') with \(|G:H_p(G)|=p^2\) for \(p=5\). Later the reviewer [Mat. Sb., N. Ser. 116(158), 253-264 (1981; Zbl 0473.20016)] showed that such an anti-Hughes \(p\)-group exists whenever a certain Wall relation \(W_{2p-1}\) in the associated Lie ring \(L(B(p))\) of the free countably generator group of exponent \(p\) is not a consequence of the \((p-1)\)-Engel relation mod \(p\). Furthermore, the reviewer [Mat. Sb., N. Ser. 118(160), 567-575 (1982; Zbl 0484.20017)] showed that even a 2-generator counterexample (giving answers to a series of other questions about finite \(p\)-groups) exists if another Wall relation \(W_{2p}\) is ``new'' in the same sense. The reviewer (ibid.) used computer-aided calculations to verify that this is the case for \(p=7\), thus constructing a 2-generator anti-Hughes group for \(p=7\). In this paper the authors use computer-aided calculations to produce explicit power-commutator presentations of some 3- and 2-generator anti-Hughes groups with \(|G:H_p(G)|=p^2\), including relatively small examples with orders \(5^{46}\) and \(7^{66}\). Other calculations in \(L(B(p))\) confirm that \(W_{2p-1}\) is ``new'' for \(p=11\), 13, 17, 19, thus giving the existence of 3-generator anti-Hughes groups for these primes. (It is of course expected that this is so for all primes larger than 3.) They also produce a 2-generator anti-Hughes g…
reviewed by: Evgenii I. Khukhro
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F45
zbMATH DE Number: 5610505
zbMATH Keywords: Hughes conjecture
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: counterexamples
zbMATH Keywords: power-commutator presentations
zbMATH Keywords: Lie rings
Wikidata QID: Q123356416
describes a project that uses: Magma
describes a project that uses: GAP
MaRDI profile type: Publication
OpenAlex ID: W2165992057
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DOI: 10.1016/J.JALGEBRA.2009.04.011
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description: scientific article; zbMATH DE number 5610505