nLab
Steiner system

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Steiner systems are combinatorial objects whose main applications are in finite geometries and finite group theory.

Given positive integers l,n,ml,n,m a Steiner system of type S(l,n,m)S(l,n,m) is a pair of a set SS of cardinality nn and a set of subsets of SS of cardinality mm, called blocks, such that every ll-element subset of SS is in precisely one block.

Nontrivial Steiner systems occur for 1<l<n<m1\lt l\lt n\lt m.

Special cases are the S(2,3,n)S(2,3,n), called Steiner triple systems, and the S(3,4,n)S(3,4,n), called Steiner quadruple systems.

The simple sporadic Mathieu groups arise as automorphism groups of certain Steiner systems. For instance, the largest, M 24M_24, is composed by the automorphisms of S(5,8,24)S(5,8,24).

Last revised on August 27, 2019 at 08:23:03. See the history of this page for a list of all contributions to it.