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Let V be a Euclidean space. The finite reflection groups on V play a central role in the study of finite groups and of algebraic groups. We shall begin by classifying all the finite subgroup of the orthogonal group O(V) when V has dimension 2 or 3. For G a finite subgroup of O(V), we then introduce fundamental regions for the action of G on V. Following this we define Coxeter groups as finite groups generated by reflections in O(V) which act effectively on V. To study such subgroups of O(V) we introduce root systems and show that G simply transitively on the positive systems in the root system. In the final chapter, we classify root systems and thus also classify the Coxeter groups. This classification is as usual parameterized by the Coxeter diagrams. This classification is as usual parameterized by the Coxeter diagrams. As time allows I will cover further material. This will be chosen from: Presentations of Coxeter Groups, Invariants of Coxeter Groups, Affine Reflection groups, Complex reflection groups.


Spring 2010 (Monday, January 11 to Friday, March 19)


  • Mon 11:05 - 11:55
  • Fri 11:05 - 11:55


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Christopher Parker
Phone (0121) 414 6199
Interests Finite groups
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Photo of PEDEN Andrew
PEDEN Andrew
Photo of Heather BURKE
Heather BURKE
Photo of Tim Crinion
Tim Crinion
Photo of Asma Ismail
Asma Ismail
Photo of Anton Izosimov
Anton Izosimov
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Daniel Kirk
Photo of Parsons Mark
Parsons Mark
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John McLeod
Photo of Athirah MohamedNawawi
Athirah MohamedNawawi
Photo of Andrew Monaghan
Andrew Monaghan
Photo of Yann Palu
Yann Palu
Photo of Heather Riley
Heather Riley


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Finite Reflection Groups2.pdf

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