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CATEGORIES:Mordell Lectures
SUMMARY:Short presentations of finite groups - Alex Lubotz
ky
DTSTART;TZID=Europe/London:20110428T170000
DTEND;TZID=Europe/London:20110428T183000
UID:TALK28061AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/28061
DESCRIPTION:Finding 'nice & compact' presentations of various
groups has been a subject of much research for mor
e than a century. The Coxeter presentation of the
symmetric groups and the Steinberg presentation of
groups of Lie type are such. In response to conje
ctures of Babai and Szemeredi on one hand (motivat
ed by questions in computational group theory) and
of Mann on the other hand (motivated by questions
on subgroup growth) we show that all non-abelian
finite simple groups (with the possible exception
of Ree groups) have presentations which are small
(bounded number of relations) and short (w.r.t the
length of the relations). This is surprising as t
he simple abelian groups - the cyclic groups of pr
ime order - do not have such presentations! We wil
l describe the motivations and results\, a cohomol
ogical application (proving a conjecture of Holt)
and some connections with discrete subgroups of Li
e groups and topology.\n\nThe talk will be suitabl
e for undergraduates with basic knowledge of group
theory.\n\nThe talk is based on a series of paper
s with Bob Guralnick\, Bill Kantor and Martin Kass
abov.\n\n 1. Presentations of finite simple grou
ps: profinite and cohomological approaches\, Group
s\, Geometry and Dynamics 1 (2007) 469-523.\n 2.
Presentations of finite simple groups: a quantita
tive approach\, J. of the AMS 21 (2008) 711-774.\n
3. Presentations of finite simple groups: a com
putational approach\, J. of the Euro. Math. Soc. 2
(2011) 391- 458.
LOCATION:Wolfson Room (MR 2) Centre for Mathematical Scienc
es\, Wilberforce Road\, Cambridge
CONTACT:
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