BEGIN:VCALENDAR VERSION:2.0 PRODID:-//talks.bham.ac.uk//v3//EN BEGIN:VEVENT CATEGORIES:Algebra seminar SUMMARY:Permutation puzzles and finite simple groups - Jas on Semeraro\, University of Bristol DTSTART:20141023T150000Z DTEND:20141023T160000Z UID:TALK2585AT URL:/talk/index/2585 DESCRIPTION: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 appropria te sequence of moves. The set of sequences of move s which leave the hole in a fixed position forms a finite group (the puzzle group) which is easily s een to be isomorphic to the alternating group Alt( 15).\nVarious generalisations of the 15-puzzle hav e already been studied. For example\, Wilson consi ders an analogue for finite connected and non-sepa rable graphs. More recently\, Conway introduced a version of the puzzle which is played with counter s on 12 of the 13 points in the finite projective plane P(3). The 13th point _h_ (called the hole) m ay be interchanged with a counter on any other poi nt _p_\, provided the two counters on the unique l ine containing _h_ and _p_\nare also interchanged. It turns out that the group of move sequences whi ch fix the hole is isomorphic to the Mathieu group M12.\nIn this talk\, we extend Conway' s game to arbitrary simple 2-(_n_\,4\,λ)-designs w ith the property that any two lines intersect in a t most two points. We obtain a plethora of example s of puzzle groups including the symplectic and or thogonal groups in characteristic 2. We completely classify puzzle groups when λ<3\, and show that t here are finitely many isomorphism classes of puzz le groups for each λ>0. We also apply Mihailescu’s theorem (formally Catalan’s conjecture)\nto give a new characterisation of M12.\n LOCATION:Physics West 117 CONTACT:David Craven END:VEVENT END:VCALENDAR