A Steiner system 
is a set 
of 
points, and a set
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 an 
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
 |
For 
,
these numbers also satisfy 
and 
by Fisher's
block design inequality.
The permutations of the points preserving the blocks of a Steiner system 
form the automorphism group
of 
.
For example, consider 
the set of 9 points in the two-dimensional vector space
over the field of three elements.
The blocks are the 12 lines of the form 
with 
, which have three elements
each. The system is an 
because any two points uniquely determine a line.
For this construction in affine geometry, the affine transformations preserving the lines
form a subgroup of the automorphism
group. For a vector space of dimension 
over a field
of 
elements, the same construction gives a Steiner system 
.
Several interesting groups arise as automorphism groups of Steiner systems. The following table lists the Mathieu
groups associated with the Witt designs. In the 
case, 
has index 2 in the
full automorphism group 
; the other listed groups
are the full automorphism groups. These groups are unique up to isomorphism,
and are not only sporadic simple
groups, but are also highly transitive.
| Mathieu group | Steiner system |
 |  |
 |  |
 |  |
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