Permutation puzzles and finite simple groups

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• Jason Semeraro, University of Bristol
• Thursday 23 October 2014, 16:00-17:00
• Physics West 117.

If you have a question about this talk, please contact David Craven.

The 15 puzzle is a sliding puzzle that consists of a frame of square tiles in random order with one tile missing (the hole), and where the aim is to obtain an ordered arrangement through an appropriate sequence of moves. The set of sequences of moves which leave the hole in a fixed position forms a finite group (the puzzle group) which is easily seen to be isomorphic to the alternating group Alt(15). Various generalisations of the 15-puzzle have already been studied. For example, Wilson considers an analogue for finite connected and non-separable graphs. More recently, Conway introduced a version of the puzzle which is played with counters on 12 of the 13 points in the finite projective plane P(3). The 13th point h (called the hole) may be interchanged with a counter on any other point p, provided the two counters on the unique line containing h and p are also interchanged. It turns out that the group of move sequences which fix the hole is isomorphic to the Mathieu group M₁₂. In this talk, we extend Conway’s game to arbitrary simple 2-(n,4,λ)-designs with the property that any two lines intersect in at most two points. We obtain a plethora of examples of puzzle groups including the symplectic and orthogonal groups in characteristic 2. We completely classify puzzle groups when λ 0. We also apply Mihailescu’s theorem (formally Catalan’s conjecture) to give a new characterisation of M₁₂.

This talk is part of the Algebra seminar series.

This talk is included in these lists:

• Algebra seminar
• Physics West 117
• School of Mathematics events

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