Steiner System (original) (raw)
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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.