Symmetric 1-factorizations of the complete graph

Let S(2n) be the symmetric group of degree 2n. We give a strong indication to prove the existence of a 1-factorization of the complete graph on (2n)! vertices admitting S(2n) as an automorphism group acting sharply transitively on the vertices. In particular we solve the problem when the symmetric group acts on 2p 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. (C) 2009 Elsevier Ltd. All rights reserved.