On counterexamples to the Hughes conjecture

AbstractIn 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define Hp(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 Hp(G)=1, Hp(G)=G, or [G:Hp(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