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
