Disjoint Steiner systems associated with the Mathieu groups

A Steiner system of type t-d-n is a collection, 3D, of subsets of a set 5 satisfying: (i) The cardinality of S is n. (ii) Each subset in 3D has cardinality d. (iii) Every subset of S of cardinality t is contained in precisely one subset in 3D. Here t, d, and n are positive integers satisfying t<d<n. Two Steiner systems, 5, 3D and S', 3D' are called equivalent if there is a bijection, <j>\ S—»S', such that <t>(D) £ 3D' if and only if P £ 3 ) . If 5 = S' and 3D = 3D', the set of equivalences forms a group, called the automorphism group of the Steiner system 5, 3D; it is a subgroup of the symmetric group on 5. If 5, 3D and 5, 3D' are Steiner systems of the same type they are said to be disjoint if 3DH3)' is empty. Among the most remarkable Steiner systems are the five associated with the Mathieu groups; i.e., those five Steiner systems whose automorphism groups are the five Mathieu groups. Witt [5] and [6] discussed them in detail, showing that they were unique up to equivalence. Two of these five systems are of central importance. For if 5, 3D is a Steiner system of type t-d-n and X a subset of S of cardinality h with h<t, then S X , {Dr\(S-X)\XC-D£3D} is a Steiner system of type (t — h)-(d — h)-(n — h). The two central Steiner systems referred to are of types 5-6-12 and 5-8-24. Their automorphism groups are, respectively, the Mathieu groups Mu and Af24. The other three Steiner systems, of types 4-5-11, 4-7-23, and 3-6-22, are derived from these two by the above method ; their automorphism groups are respectively, Mm M%%, and M22. We have found a simple proof of the following