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arXiv · 1405.1701

Generating groups using hypergraphs

Nick Gill·Neil I. Gillespie·Anthony Nixon·Jason Semeraro
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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 $\lambda \leq 2$.

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Nick Gill, Neil I. Gillespie, Anthony Nixon, Jason Semeraro. 2014-05-07. Generating groups using hypergraphs. https://arxiv.org/abs/1405.1701

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