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We consider one-factorizations of K 2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one-factors which are ®xed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non-existence statements in case the number of ®xed one-factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given.

Let F be a 2-factorization of the complete graph K v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex-set V (K v ) can then be identified with the point-set of AG(n, p) and each 2-factor of F is the union of p-cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of AGL(n, p) in this case. The proof relies on the classification of 2-(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously.

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