Math 6270 Fall 2019
## MATH 6270: Theory of Groups (Fall 2019)

Syllabus
### Schedule

- 08/26: revision of group actions, Fundamental Counting Principle for orbit size, Lagrange's Theorem
- 08/28: centralizer, normalizer, center of a p-group, existence of Sylow subgroups
- 08/30: Sylow's Theorem, Cauchy's Theorem, free groups, universal mapping property
- 09/04: construction and uniqueness of free groups, rank
- 09/06: presentations, finitely presented groups, word problem
- 09/09: verbal subgroups, varieties and HSP
- 09/11: free groups in varieties, Burnside problem
- 09/13: subnormal series, simple groups, extensions, semidirect product, solvable groups
- 09/16: commutators, derived series, central series, nilpotent groups
- 09/18: upper and lower central series, unitriangular groups
- 09/20: characterizations of nilpotent groups, Fitting subgroup controlling structure of solvable group
- 09/23: Frattini subgroup

### Handouts

- Group actions, Sylow's Theorem [pdf]

### Assignments

- due 09/04 [pdf]
- due 09/11 [pdf]
- due 09/18 [pdf]
- due 09/25 [pdf]

### Reading

The following books are on reserve in Gemmill library.
- I. Martin Isaacs. Finite group theory. AMS, 2008.
- Derek J.S. Robinson. A course in the theory of groups. Springer, 2nd edition, 1996.

You can freely download the pdf for the book from Springer through our library via
this link
- Joseph J. Rotman. An introduction to the theory of groups. Boston : Allyn and Bacon, 3rd edition, 1984.