M390C: Finite Group Theory (graduate course)
Daniel Allcock, RLM 9.112, phone 1-1120
(my last name)@math.utexas.edu.

The transfer homomorphism: Burnside's, Grün's and Frobenius' theorems
Fusion: Alperin fusion theorem; role of tame intersections
Extensions: basic formalism, Schur-Zassenhaus theorem, existence of Schur covering group, computation for various groups including alternating groups and PSL₂(q).
Basic representation and character theory: as in Dummit-Foote chs. 18-19
Fancier character theory: Frobenius' theorem, exceptional characters, Brauer-Suzuki theorem, characterization of PSL₂(2ⁿ), CA-groups of odd order
Chevalley groups of types A, D, and E: construction, important subgroups, simplicity, fusion in Sylow subgroups in defining characeristic, and the Steinberg presentation and Schur multiplier

Of course, if you are reading this then you probably don't know what any of these are. So, more verbosely: the transfer homomorphism lets you find abelian quotients of a finite group G from purely "local" constructions, where local means "having to do with the Sylow p-subgroups for one particular prime p". Fusion means understanding when two elements of a Sylow subgroup, not conjugate in that subgroup, become conjugate in G. The Schur-Zassenhaus theorem shows that if a normal subgroup of G has order relatively prime to the quotient by it, then it has a complement and G is a semidirect product. The Schur covering group is the universal central extension of G (under a minor hypothesis on G), and knowing its center is part of understanding G.

The character theory beyond that in Dummit-Foote gives several amazing theorems fairly easily. The Brauer-Suzuki theorem gives strong information about an arbitrary finite group G whose 2-Sylow subgroups are quaternion groups, in particular that G is not simple. Suzuki's theorem on CA-groups is that a group of odd order is solvable if every element has abelian centralizer. It is a baby case of the Feit-Thompson theorem that all groups of odd order are solvable.

The Chevalley groups are the analogues over arbitrary fields of Lie groups. We will develop them without any algebraic-geometry machinery, and see what the major constructions in the course mean for these groups. The restriction to ADE is to make the ideas as transparent as possible.

Also, if my students want a particular topic then I can be sure to cover that. I feel an obligation to cover the Thompson subgroup of a p-group, and show how to work with my favorite sporadic simple group (M₂₄, of order 244,823,040). I will if I have time but I probably won't.

I will distribute notes for the course, except for the material in Dummit-Foote. My main sources are Isaacs' "Finite Group Theory", Gorenstein's "Finite Groups", Collins' "Representations and Characters of Finite Groups" and Steinberg's mimeographed notes on Chevalley groups.

PREREQUISITES: Thorough understanding of the group theory material from the first-semester prelim course. Some first-year students will be fully able to take the course, but those who are just skating by will not.

GRADING: for students advanced to candidacy, grading is largely meaningless. Students not yet advanced to candidacy will have homework to do (substantial but less than a prelim course) and also a presentation and/or a short paper to write.