| Achoimre: | <p>In 1957 D.R. Hughes published the following problem in group theory. Let <em>G</em> be a group and <em>p</em> a prime. Define <em>H<sub>p</sub></em>(<em>G</em>) to be the subgroup of <em>G</em> generated by all the elements of <em>G</em> which do not have order <em>p</em>. Is the following conjecture true: either <em>H<sub>p</sub></em>(<em>G</em>)=1, <em>H<sub>p</sub></em>(<em>G</em>)=<em>G</em>, or [<em>G</em>:<em>H<sub>p</sub></em>(<em>G</em>)]=<em>p</em>? After various classes of groups were shown to satisfy the conjecture, G.E. Wall and E.I. Khukhro described counterexamples for <em>p</em>=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<sup>46</sup> and 7<sup>66</sup>. It is expected that the conjecture is false for all primes larger than 3. We show that it is false for <em>p</em>=13,17 and 19.</p>
|
|---|