On counterexamples to the Hughes conjecture

Journal article

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)=G, or [G:H_p(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 5⁴⁶ and 7⁶⁶. 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.

Authors: Havas, G; Vaughan-Lee, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Algebra
• Volume: 322
• Issue: 3
• Pages: 791-801
• Publication date: 2009-08-01
• Edition: Publisher's version
• DOI: 10.1016/j.jalgebra.2009.04.011
• ISSN: 0021-8693
• Language: English
• Keywords: p-groups; power-commutator presentations; Lie rings; counterexamples; Hughes conjecture
• Subjects: Mathematics; Group theory and generalizations (mathematics)