august 29th - september 9th


Summer School
Finite Groups
Related Geometrical Structures

in memory of Maria Silvia Lucido (1963-2008)














Exceptional and Sporadic Groups

Course 1.

Lectures by Alexander Ivanov

1. Amalgam method for constructing groups on example of the Coxeter group of type B_k

2. The theory of extraspecial 2-groups; their classification, automorphism groups and representations.

3. Triextraspecial groups as an illustration of how to use the first and second cohomology groups and to construct ''triality'' automorphisms

4. The Mathieu groups and the relevant Steiner systems and codes.

5. Leech lattice and the Conway groups.

6. Axiomatics of the Majorana theory.

7. Sakuma-Norton algebras and their properties; Sakuma's theorem.

8. Majorana representations of S_4

9. Constructing the Monster via the amalgam method and the Y-presentation of the Monster.

10. Majorana representations of the ''standard'' A_{12} subgroup in the Monster and of its subgroups.


M. Aschbacher, Finite Group Theory,
Cambridge University Press, Cambridge 1986.

M. Aschbacher, Sporadic groups,
Cambridge University Press, Cambrdige  1994.
J. H. Conway, R. T. Curtis, R. A. Parker, S. P. Norton, R. A. Wilson - Atlas of finite groups.
Clarendon Press, Oxford 1985 (reprinted with corrections 2004)

A. A. Ivanov, Geometries  of Sporadic Groups I. Petersen and Tilde Geometries, 
Cambridge University Press, Cambridge 1999

A. A. Ivanov, S. V. Shpectorov,  Geometries  of Sporadic Groups II. Representations and Amalgams,  Cambridge University Press, Cambridge 2002.

A. A. Ivanov, $J_4$,
Oxford Univ. Press, Oxford, 2004.

A. A. Ivanov, The Monster Group and Majorana Involutions,
Cambridge Univ. Press, Cambridge 2009.

A. A. Ivanov, Constructing the Monster amalgam
J. Algebra  300 (2005), 571--589.

A. A. Ivanov, D. V. Pasechnik, A. Seress and S. Shpectorov, Majorana representations of the symmetric group of degree 4
J. Algebra 324 (2010), 2432-2463.

A. A. Ivanov and S. V. Shpectorov, Tri-extraspecial groups
J. Group Theory 8 (2005), 395--413.

Course 2.

Lectures by Robert Wilson

The first week is on exceptional groups of Lie type, but without using any Lie theory. The second week is on sporadic groups. The two weeks are largely independent of each other, except that the last lecture in week 2 uses octonions, introduced in week 1.

1. The Suzuki groups $Sz(q)$, from first principles.

2. Octonion algebras and construction of the simple groups $G_2(q)$.

3. The exceptional behaviour of $G_2(q)$ in characteristic $3$, and construction of the small Ree groups ${}^2G_2(3^{2n+1})$.

4. $D_4(q)$ and triality.

5. Exceptional Jordan algebras and $F_4(q)$.

6.  $M_{24}$ and the binary Golay code.

7. $M_{12}$ and the ternary Golay code.

8. The Leech lattice and Conway's groups.

9. Complex and quaternionic Leech lattices.

10.  The octonionic Leech lattice.


The bibliography contains references to a number of talks of mine in which various of the above topics are introduced at a (hopefully) reasonably accessible level.

R. A. Wilson, The finite simple groups, Springer GTM 251, 2009.
Especially sections 4.2, 4.3, 4.5, 4.7, 4.8, 5.2, 5.3, 5.4, 5.6.

R. A. Wilson, The Suzuki groups,

R. A. Wilson, E_8,

R. A. Wilson, Octonions,

R. A. Wilson, The Golay code,

R. A. Wilson, The Leech lattice,

R. A. Wilson, An octonionic Leech lattice,

R. A. Wilson, The weird and wonderful world of octonions,

Beginning: Monday, August 29th, 2011 at 9.30am.

End: Friday, September 9th 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. 31st, Friday 2nd, Monday 5th, and Wednesday Sep. 7st
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 8th