Generating groups using hypergraphs

Gill, Nick; Gillespie, Neil I.; Nixon, Anthony and Semeraro, Jason (2016). Generating groups using hypergraphs. The Quarterly Journal of Mathematics, 67(1) pp. 29–52.



To a set B of 4-subsets of a set Ω 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 (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2⁠.

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