Symmetry groups related to the construction of perfect one factorizations of \(K_{2n}\) (Q1113924)

Let \(X=\{1,2,...,2n\}\), S(X) be the symmetric group on X. A subset \(P\subset S(X)\) which consists of 2n-1 involutions without fixed points is called a one factorization of \(K_{2n}\). The one factorization P is called perfect if a group generated by every two of these involutions is transitive. Let S(X) act on itself by conjugations. Denote by \(I_ P\) the group of all permutations from S(X) stabilizing P. The main result: If P is a perfect one factorization and \(I_ P\) has a noncentral involution with a fixed point, then P coincides with one of the known perfect one factorizations of \(K_{p+1}\) or \(K_{2p}\) where p is prime.