nLab Steiner system

Idea

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

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

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

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

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

• Wikipedia: Steiner system