Symmetric 1-factorizations of the complete graph

Let S 2 n be the symmetric group of degree 2 n . We give a strong indication to prove the existence of a 1-factorization of the complete graph on ( 2 n ) ! vertices admitting S 2 n as an automorphism group acting sharply transitively on the vertices. In particular we solve the problem when the symmetric group acts on 2 p elements, for any prime p . This provides the first class of G -regular 1-factorizations of the complete graph where G is a non-soluble group.