Generating groups using hypergraphs

TL;DR

This paper introduces the 'hole stabilizer' invariant for 4-subset sets, generalizes a Mathieu group construction, and classifies structures with trivial or small hole stabilizers, linking combinatorial designs and group theory.

Contribution

It defines the hole stabilizer invariant, relates it to partial groups, and classifies certain combinatorial structures based on the stabilizer properties.

Findings

Classified pairs with trivial hole stabilizer.

Determined all hole stabilizers for 2-(n,4,λ) designs with λ ≤ 2.

Connected hole stabilizers to objective partial groups.

Abstract

To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd'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 $(\Omega,\mathcal{B})$ with a trivial hole stabilizer, and determine all hole stabilizers associated to $2$-$(n,4,\lambda)$ designs with $\lambda \leq 2$.