Math 6270 Fall 2019

MATH 6270: Theory of Groups (Fall 2019)

Syllabus

Schedule

  1. 08/26: revision of group actions, Fundamental Counting Principle for orbit size, Lagrange's Theorem
  2. 08/28: centralizer, normalizer, center of a p-group, existence of Sylow subgroups
  3. 08/30: Sylow's Theorem, Cauchy's Theorem, free groups, universal mapping property
  4. 09/04: construction and uniqueness of free groups, rank
  5. 09/06: presentations, finitely presented groups, word problem
  6. 09/09: verbal subgroups, varieties and HSP
  7. 09/11: free groups in varieties, Burnside problem
  8. 09/13: subnormal series, simple groups, extensions, semidirect product, solvable groups
  9. 09/16: commutators, derived series, central series, nilpotent groups
  10. 09/18: upper and lower central series, unitriangular groups
  11. 09/20: characterizations of nilpotent groups, Fitting subgroup controlling structure of solvable group
  12. 09/23: Frattini subgroup

Handouts

  1. Group actions, Sylow's Theorem [pdf]

Assignments

  1. due 09/04 [pdf]
  2. due 09/11 [pdf]
  3. due 09/18 [pdf]
  4. due 09/25 [pdf]

Reading

The following books are on reserve in Gemmill library.
  1. I. Martin Isaacs. Finite group theory. AMS, 2008.
  2. 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
  3. Joseph J. Rotman. An introduction to the theory of groups. Boston : Allyn and Bacon, 3rd edition, 1984.