Simple Groups

Gernot Stroth

Syllabus: The course consists of two main parts.

A.  Sporadic Groups 

1. Understanding and constructing the Mathieu Groups.
2. The group J₂:  Starting with the centralizer of an involution, we determine the order of J₂. On the way we will develop basic tools like group order formulas, transfer and fusion.
3. We will construct the simple group HiS.

B. p-local theory

We shall develop important tools for studying p-local subgroups in finite groups (mostly p=2).

1. Quadratic groups in characteristic two: Basic properties, why Lie groups in odd characteristic usually do not possess quadratic modules in characteristic two, construction of modules using ideas from Sheaf Homology. As an example we shall construct the 12-dimensional module for 3U₄(2) and maybe some module for a sporadic group.
2. F-modules and 2F-modules. How such modules arise and how they can be used to determine the p-local structure of a group. As an example we shall construct the p-local structure of a group of Lie type and a sporadic group. 

References:

1. M. Aschbacher, Finite Group Theory, Cambridge University press 1986.

2. N. Blackburn and B. Huppert, Finite Groups III, Springer 1982.

3. H. Bender, Steiner systems and the Mathieu groups revisited, Groups and Combinatorics - in memory of Michio Suzuki,  Math. Soc. Japan, (2001), 255-278.

4. D. Gorenstein, Finite Simple Groups, Plenum 1985.

5. G. Highman, M. Powell (eds.) Finite Simple Groups, Academic Press 1971.

6. A. A. Ivanov, Geometry of Sporadic Groups, Cambridge University Press 1999. 

7. H. Kurzweil, B. Stellmacher, The Theory of Finite Groups, an Introduction, Springer 2004.

8. U. Meierfrankenfeld, G. Stroth, Quadratic GF(2)-modules for sporadic simple groups, Comm. in Algebra 18 (1990), 2099-2140.

9  U. Meierfrankenfeld, G. Stroth, On quadratic GF(2)-modules for Chevalley groups over fields of odd order, Arch. Math. 55 (1990), 105-110.

10.  U. Meierfrankenfeld, G. Stroth, The H-structure Theorem, in preparation.

11. M. Ronan, S. Smith, Computation of 2-modular sheaves and representations for L₄(2), A₇, 3S₆, and M₂₄, Comm. in Algebra 17 (1989), 1199-1237.

12 M. Suzuki,  Group Theory II, Springer 1986.