GROUP THEORY

Brief history of the subject

Instructor: Prof. Gregory M. Constantine; 509 TY, 624-8308

Texts: Dummit and Foote's text on Algebra and Knapp's "Basic Algebra". See also Alperin and Bell's "Groups and representations", Springer,1998, and Serre's "Linear representations of finite groups", also a Springer publication.

Office hours: Wednesdays and Fridays 11 - 11:50 and by appointment.

Grades: Homework 30%, Exam 1 20%, Exam 2 20%, Final exam 30%

Homework assigned during a week is due Friday of the following week.

COURSE OUTLINE

Part 1: Group actions
SUBJECT MATTER 1

Week 1 Definitions and Examples

Examples include the cyclic group, groups of linear maps, symmetric, alternating, dihedral groups, and groups of symmetries of the Platonic solids.

Week 2

1.Lagrange's Theorem:

The order of a subgroup divides the order of the group.

2.The class equation:

The length of an orbit equals the index of the stabilizer.

Cayley's Theorem: Every group can be represented as a regular permutation group.

3.Frobenius-Cauchy Lemma:

The number of orbits equals the average number of points fixed by elements of the group.

4.Polya and DeBruijn's theorems on counting in the presence of groups.

5. Simplicity of the alternating groups.

Week 3

6.Sylow's Theorem:

The Sylow p-subgroups are equal to 1 mod p in number and they are all conjugate in the group.

7.Frobenius' Theorem:

If n divides the order of the group, then the number of elements whose order divides n is a multiple on n.

Exam 1

Part 2: Abelian, Solvable, and Nilpotent groups
SUBJECT MATTER 2

Week 4

a. The canonical decomposition of an Abelian group into cyclic subgroups.

b. Delsarte's result on automorphisms of Abelian groups

Week 5

c. The Jordan- Holder theorem:

Any two composition series of a group are equivalent.

d. Basic results on solvable groups.

Week 6

e. Fitting Lemma; The Krull- Schmidt Theorem

f. Hall's Anzahl Theorems on p-groups

Part 3: Linear representations
SUBJECT MATTER 3

Week 7

Th1: Each representation is a direct sum of irreducibles.

Th2:Schur's lemma with corollaries.

Week 8

Th3: Character tables.

Th4: Canonical decomposition of a representation.

Week 9

Th5: Induced representations.

Week 10

Proofs of Frobenius' reciprocity law, Mackey's criterion on irreducibility, Burnside's PaQb theorm, the existence of Frobenius kernels, and Artin and Brauer results on integrality of characters induced from cyclic and elementary subgroups. [This material is more specialized and may be trimmed to fit the interests of the audience.]

Exam 2

Part 4: Simple groups
SUBJECT MATTER 4

Week 11

The general, special, and projective linear groups.

Dickson's Theorem:

The group PSL(n,q) is simple.

Week 12

Steiner systems, affine and projective geometries.

Week 13

The Mathieu groups

Week 14

Connections to Coding theory and Statistical design

Week 15

Connections to Computer science and Probability

Week 16 Review; Final exam

Recommended reading:

Gorenstein's "Finite groups"

Curtis and Reiner's "Representation theory of finite groups and associative algebras", Interscience Pub., New York, 1962

Feit's "Characters of finite groups", Benjamin Pub., New York, 1967

Here are some oldies but goodies:

Burnside's turn of the century treatice

Weyl's "The theory of groups and quantum mechanics", 1931, Dover

Miller, Blichfeldt, and Dickson's "Theory and applications of finite groups", 1916, also 1961 Dover

Carmichael's "Groups of finite order", 1937, also 1956 Dover

Exercises:

Dixon's "Problems in group theory", 1973 Dover

[I'll add more to this list.]

Enjoy!