Generating minimally transitive permutation groups. (Q296672)

A transitive permutation group \(G\leq S_n\) is called minimally transitive if every proper subgroup of \(G\) is intransitive. The author, answering an old question proposed by L. Pyber, proves that each minimally transitive permutation group of degree \(n\) can be generated by \(\mu(n)+1\) elements, where \(\mu(n)\) is the largest positive integer \(m\) for which there exists a prime \(p\) such that \(p^m\) divides \(n\). The proof uses a method to approach the question of finding the minimal number of elements required to generate a finite group, which is due to F. Dalla Volta and the reviewer and is based on the properties of the crown-based powers.