On counterexamples to the Hughes conjecture

JOURNAL OF ALGEBRA

卷:322

Article

Havas, George¹  Vaughan-Lee, Michael² 

[1] Univ Queensland, Sch Informat Technol & Elect Engn, ARC Ctr Complex Syst, Brisbane, Qld 4072, Australia

[2] Univ Oxford Christ Church, Oxford OX1 1DP, England

关键词: Hughes conjecture;    p-Groups;    Counterexamples;    Power-commutator presentations;    Lie rings;   

DOI  :  10.1016/j.jalgebra.2009.04.011

来源: Elsevier

【 摘 要 】

In 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define H-p(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-p(G) = 1, H-p(G) = C, or vertical bar G : H-p(G)vertical bar = 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(46) and 7(66). 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. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free