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, Frattini argument, nilpotence of finite Frattini subgroup
  13. 09/25: Hall pi-subgroups, crossed homomorphisms
  14. 09/27: Schur-Zassenhaus Theorem, existence and conjugacy of complements
  15. 09/30: pi-separable groups, existence of Hall pi-subgroups in solvable groups
  16. 10/02: conjugacy of Hall pi-subgroups in solvable groups, transfer homomorphism
  17. 10/04: transfer evaluation lemma, transfer into center
  18. 10/07: transfer into abelian Sylow subgroups
  19. 10/09: Burnside's normal p-complement theorem, groups with all Sylow subgroups cyclic
  20. 10/11: group algebras, linear representations of algebras and groups
  21. 10/14: Maschke's Theorem, Schur's Lemma, irreducible representations of abelian groups
  22. 10/16: Wedderburn Artin Theorem for semisimple algebras, structure of group algebras over algebraically closed fields
  23. 10/18: representations of S_3, characters
  24. 10/21: irreducible characters as basis for class functions over the complex numbers, decomposition of the regular character and of 1
  25. 10/23: orthogonality relations for characters
  26. 10/25: character tables
  27. 10/28: product of characters via tensor products of modules, characters of products of groups
  28. 10/30: characters are algebraic integers
  29. 11/01: irreducible character degree divides the group order
  30. 11/04: nontrivial conjugacy classes in nonabelian simple groups do not have prime power size, Burnside's p^aq^b-Theorem
  31. 11/06: induced characters, Frobenius reciprocity, Frobenius group defined by complement
  32. 11/08: existence of Frobenius kernel
  33. 11/11: group extensions, short exact sequences, equivalence between sequences yield isomorphism between extensions
  34. 11/13: split extensions, couplings, G-modules, extensions determine factor sets (2-cocyles), Z^2(G,A)
  35. 11/15: 2-coboundaries B^2(G,A), factor sets determine extensions, equivalence classes of extensions are in bijection with H^2(G,A) = Z^2(G,A)/B^2(G,A)
  36. 11/18: cochain complexes, projective resolutions, Ext^n(M,D) is independent of choice of projective resolution of M (Dummit-Foote 17.1 Thm 6)
  37. 11/20: standard resolution of the trivial ZG-module Z, n-cocyles Z^n(G,M), n-coboundaries B^n(G,M) and their quotient H^n(G,M) (n-th cohomology group of G with coefficients in M)

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]
  5. due 10/02 [pdf]
  6. due 10/09 [pdf]
  7. due 10/16 [pdf]
  8. due 10/23 [pdf]
  9. due 10/30 [pdf]
  10. due 11/08 [pdf]
  11. due 11/13 [pdf]
  12. due 11/20 [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.