Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements

For a finite group G, the set of all prime divisors of |G| is denoted by pi(G). P. Shumyatsky introduced the following conjecture, which was included in the "Kourovka Notebook" as Question 17.125: a finite group G always contains a pair of conjugate elements a and b such that pi(G) = pi(aOE (c) a, b >). Denote by the class of all finite groups G such that pi(H) not equal pi(G) for every maximal subgroup H in G. Shumyatsky's conjecture is equivalent to the following conjecture: every group from is generated by two conjugate elements. Let be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that . We prove that every group from is generated by two conjugate elements. Thus, Shumyatsky's conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatsky's conjecture.