august 23rd - september 3rd


Summer School
Finite Groups
Related Geometrical Structures

in memory of Maria Silvia Lucido (1963-2008)














Permutation Groups and Applications.

Permutation group theory is a classical part of Group theory, going back to Galois, Jordan and others. The face of the subject changed considerably with the classification of finite simple group.
The course will give some classical results, then outline the modern approach, followed by applications to graphs, geometries, and probabilistic group theory.

Course 1

Lectures by Jan Saxl

1.  G-spaces; primitivity and imprimitivity; wreath products
2.  Doubly transitive groups; Burnside reduction theorem; Jordan groups;
3.  Simply primitive groups; orbital graphs; double cosets; permutation
    characters; centraliser rings
4.  Symmetric groups and their subgroups; maximality and factorizations
5.  Classical groups and their actions; natural subgroups
6.  Permutation groups of special degrees
7.  Factorizations of groups
8.  Flag-transitive linear spaces
9.  Distance-transitive graphs
10. Gelfand pairs

Course 2

1st week

1.  Generalized Fitting Subgroup (Mario Mainardis)
2.  Aschbacher-O'Nan-Scott Theorem (John van Bon)
3.  Introduction to Derangements and fixed point ratios (John Britnell)
4.  A brief introduction to groups of Lie type (Andrea Previtali)
5.  Maximal subgroups of classical groups - Aschbacher's theorem (John van Bon)

2nd week

Lectures by Martin Liebeck

6. Fixed points
7. Monodromy groups
8. Bases for permutation groups
9. Derangements with applications to polynomials in characteristic p
10. Probabilistic aspects

Some references:

PJ Cameron, Permutation groups, LMS Students Texts 45, CUP 1999.
JD Dixon and B Mortimer, Permutation groups, Springer-Verlag 1996.
P Kleidman and M Liebeck, The subgroup structure of the finite classical
  groups, LMS Lecture Note Series 129, CUP 1990.
RM Guralnick, Some applications of subgroup structure to probabilistic
  generation and covers of curves, in Algebraic groups and their
  representations (Cambridge, 1997),  eds RW Carter and J Saxl, 301-320,
  NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ. 1998.
RM Guralnick, Monodromy groups of coverings of curves, Galois groups and
  fundamental Monodromy groups of coverings of curves, in Galois groups
  and fundamental groups, ed L Schneps, 1--46, Math. Sci. Res. Inst. Publ.
  41, CUP 2003.
G Malle and D Testerman, A course on linear algebraic groups and finite groups
  of Lie type, Lecture notes Venice 2007, to appear.
H Wielandt, Finite permutation groups, Academic Press 1964
RA Wilson, The finite simple groups, Graduate Texts in Mathematics 251,
  Springer-Verlag 2009.

Beginning: Monday, August 23rd, 2010 at 9.30am.

End: Friday, September 3rd 2010 at 1.00pm.

Day schedule:

  9.30am - 11.00am: Lecture 1
11.00am - 11.30am: Break
11.30am -   1.00pm: Lecture 2

Class problem sections on Wednesday Aug. 25th, Friday 27th, Monday 30th, and Wednesday Sep. 1st
with the following schedule:

15.30pm - 16.30pm: Class problem section 1
16.30pm - 17.00pm: Break
17.00pm - 18.00pm: Class problem section 2

Course Banquet on Thursday, September 2nd