On counterexamples to the Hughes conjecture

Havas, George and Vaughan-Lee, Michael (2009) On counterexamples to the Hughes conjecture. Journal of Algebra, 322 3: 791-801. doi:10.1016/j.jalgebra.2009.04.011

Author Havas, George
Vaughan-Lee, Michael
Title On counterexamples to the Hughes conjecture
Journal name Journal of Algebra   Check publisher's open access policy
ISSN 0021-8693
Publication date 2009-08-01
Year available 2009
Sub-type Article (original research)
DOI 10.1016/j.jalgebra.2009.04.011
Open Access Status
Volume 322
Issue 3
Start page 791
End page 801
Total pages 11
Editor Michel Broue
Gerhard Hiss
Place of publication Amsterdam , The Netherlands
Publisher Elsevier
Language eng
Subject 010105 Group Theory and Generalisations
970101 Expanding Knowledge in the Mathematical Sciences
Abstract In 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define H (G) to be the subgroup of G generated by all the elements of G which do not have order p. Is the following conjecture true: either H (G) = 1, H (G) = G, or [G : H (G)] = p? After various classes of groups were shown to satisfy the conjecture, G.E. Wall and E.I. Khukhro described counterexamples for p = 5, 7 and 11. Finite groups which do not satisfy the conjecture, anti-Hughes groups, have interesting properties. We give explicit constructions of a number of anti-Hughes groups via power-commutator presentations, including relatively small examples with orders 5 and 7. It is expected that the conjecture is false for all primes larger than 3. We show that it is false for p = 13, 17 and 19.
Keyword Hughes conjecture
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: 2010 Higher Education Research Data Collection
School of Information Technology and Electrical Engineering Publications
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Created: Thu, 03 Sep 2009, 17:57:13 EST by Mr Andrew Martlew on behalf of School of Information Technol and Elec Engineering