On counterexamples to the Hughes conjecture

In 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? This conjecture has become known as the Hughes conjecture. 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, antiHughes 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.