Generating groups using hypergraphs (Q2794437)

Some years ago, \textit{J. H. Conway} et al. [Exp. Math. 15, No. 2, 223--236 (2006; Zbl 1112.20003)] studied a construction by \textit{J. H. Conway} [Lond. Math. Soc. Lect. Note Ser. 241, 1--11 (1997; Zbl 0887.05002)] for the Mathieu group M\(_{12}\), reminiscent of the classic \(15\)-puzzle played with tiles on a \(4 \times 4\) grid. In this construction, the grid was replaced by the projective plane P\(_3\), and the \(15\) tiles replaced by \(12\) counters which sit on all but one of the \(13\) points of P\(_3\).NEWLINENEWLINEIn the current paper, the authors develop and analyse a generalisation of this structure, replacing P\(_3\) by a \(4\)-hypergraph, that is, by a pair \({\mathcal D} = (\Omega, {\mathcal B})\), where \(\Omega\) is a finite set and \({\mathcal B}\) is a finite multiset of subsets of \(\Omega\) of size 4. The subset of Sym\((\Omega)\) consisting of all move sequences for the resulting puzzle has the structure of a partial group (in the sense of \textit{A. Chermak} [Acta Math. 211, No. 1, 47--139 (2013; Zbl 1295.20021)]). The authors also give some applications for \(2\)-\((n,4,\lambda)\) designs, and a number of open questions and avenues for future work.