Generating groups using hypergraphs

To a set B of 4-subsets of a set of size n, we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd’s ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n, 4, λ) designs with λ 2.