A Steiner system is a set of points, and a collection of subsets of of size (called blocks), such that any points of are in exactly one of the blocks. The special case and corresponds to a so-called Steiner triple system. For a projective plane, , , , and the blocks are simply lines.
The number of blocks containing a point in a Steiner system is independent of the point. In fact,
where is a binomial coefficient. The total number of blocks is also determined and is given by
These numbers also satisfy and .
The permutations of the points preserving the blocks of a Steiner system is the automorphism group of . For example, consider the set of 9 points in the two-dimensional vector space over the field over three elements. The blocks are the 12 lines of the form , which have three elements each. The system is a because any two points uniquely determine a line.
The automorphism group of a Steiner system is the affine group which preserves the lines. For a vector space of dimension over a field of elements, this construction gives a Steiner system .
Several interesting groups arise as automorphism groups of Steiner systems. For example, the Mathieu groups are the automorphism groups of Steiner systems, as summarized in the following table. These groups are unique up to isomorphism, and are not only sporadic simple groups, but are also highly transitive.
Mathieu group | Steiner system |