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Steiner System


A Steiner system S(t,k,v) is a set X of v points, and a collection of subsets of X of size k (called blocks), such that any t points of X are in exactly one of the blocks. The special case t=2 and k=3 corresponds to a so-called Steiner triple system. For a projective plane, v=n^2+n+1, k=n+1, t=2, and the blocks are simply lines.

The number r of blocks containing a point in a S(t,k,v) Steiner system is independent of the point. In fact,

 r=((v-1; t-1))/((k-1; t-1)),

where (n; k) is a binomial coefficient. The total number of blocks b is also determined and is given by

 b=(vr)/k.

These numbers also satisfy v<=b and k<=r.

SteinerSystem

The permutations of the points preserving the blocks of a Steiner system S is the automorphism group of S. For example, consider Omega 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 {a+tb}={a,a+b,a+2b}, which have three elements each. The system is a S(2,3,9) 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 n over a field of q elements, this construction gives a Steiner system S(2,q,q^d).

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 groupSteiner system
M_(11)S(4,5,11)
M_(12)S(5,6,12)
M_(22)S(3,6,22)
M_(23)S(4,7,23)
M_(24)S(5,8,24)

See also

Automorphism Group, Configuration, Mathieu Groups, Simple Group, Steiner Quadruple System, Steiner Triple System, t-Design, Transitive Group, Witt Geometry

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Steiner System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinerSystem.html

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