Quaternionic starters

Let m be an integer, m >= 2 and set n = 2^m. Let G be a non-cyclic group of order 2n admitting a cyclic subgroup of order n. We prove that G always admits a starter. Therefore, there exists a one - factorization of the complete graph on 2n vertices admitting G as an automorphism group acting sharply transitively on the vertex set. For an arbitrary even n > 2 we also show the existence of a starter in the dicyclic group of order 2n.