The Nonexistence of a Certain Steiner System

Although the automorphism group of a projective plane of order 10, if one exists, must be very small, such a plane could be the derived design of a Steiner system S(3, 12, 112) with a larger group. There are several reasons why the Frobenius group of order 56 is a promising candidate for the latter group. However, in this paper it is shown that there is no S(3, 12, 112) which is fixed by this Frobenius group. It is not known if a projective plane of order 10 exists; however, if one does exist its automorphism group must have order 1 or 3 [l, 171. A plane with such a small group is difficult to analyze. It is possible, however, that this plane arises as the derived design of a Steiner system S(3, 12, 112) having a larger group, i.e., that it is a cross section of a nicer object. The possible orders for the automorphism group of a Steiner system S(3, 12, 112) are 2', 3 * 2' or 7. 2', for 0 < i < 4. In particular this group is solvable. There are several reasons for trying to find an S(3, 12, 112) which is fixed by the Frobenius group of order 56. (i) This group is a promising candidate to try since any Desarguesian aftine plane of order n is fixed by a Frobenius group (namely, the group of order n(n-1) consisting of the mappings