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The Mathieu group \(M_{12}\) and its pseudogroup extension \(M_{13}\). (English) Zbl 1112.20003

Summary: We study a construction of the Mathieu group \(M_{12}\) using a game reminiscent of Loyd’s “15-puzzle”. The elements of \(M_{12}\) are realized as permutations on \(12\) of the \(13\) points of the finite projective plane of order \(3\). There is a natural extension to a “pseudogroup” \(M_{13}\) acting on all \(13\) points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both \(M_{12}\) and \(M_{13}\). We develop these results, and extend them to the double covers and automorphism groups of \(M_{12}\) and \(M_{13}\), using the ternary Golay code and \(12\times 12\) Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

MSC:

20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
51E20 Combinatorial structures in finite projective spaces
20D08 Simple groups: sporadic groups
94B25 Combinatorial codes
05B25 Combinatorial aspects of finite geometries
20B20 Multiply transitive finite groups