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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
M_{12}.\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 M_{12}.\n
LOCATION:Physics West 117
CONTACT:David Craven
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