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, Frattini argument, nilpotence of finite Frattini subgroup
09/25: Hall pi-subgroups, crossed homomorphisms
09/27: Schur-Zassenhaus Theorem, existence and conjugacy of complements
09/30: pi-separable groups, existence of Hall pi-subgroups in solvable groups
10/02: conjugacy of Hall pi-subgroups in solvable groups, transfer homomorphism
10/04: transfer evaluation lemma, transfer into center
10/07: transfer into abelian Sylow subgroups
10/09: Burnside's normal p-complement theorem, groups with all Sylow subgroups cyclic
10/11: group algebras, linear representations of algebras and groups
10/14: Maschke's Theorem, Schur's Lemma, irreducible representations of abelian groups
10/16: Wedderburn Artin Theorem for semisimple algebras, structure of group algebras over algebraically closed fields
10/18: representations of S_3, characters
10/21: irreducible characters as basis for class functions over the complex numbers, decomposition of the regular character and of 1
10/23: orthogonality relations for characters
10/25: character tables
10/28: product of characters via tensor products of modules, characters of products of groups
10/30: characters are algebraic integers
11/01: irreducible character degree divides the group order
11/04: nontrivial conjugacy classes in nonabelian simple groups do not have prime power size, Burnside's p^aq^b-Theorem
11/06: induced characters, Frobenius reciprocity, Frobenius group defined by complement
11/08: existence of Frobenius kernel
11/11: group extensions, short exact sequences, equivalence between sequences yield isomorphism between extensions
11/13: split extensions, couplings, G-modules, extensions determine factor sets (2-cocyles), Z^2(G,A)
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)
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)
11/20: standard resolution of the trivial ZG-module Z
11/22: 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) and their interpretations
12/02: cohomology of Z_m, H^n(G,M) is torsion for finite G
12/04: Schur-Zassenhaus Theorem (existence and conjugacy of complements for abelian kernel) via cohomology
12/06: talk by Shen Lu
12/09: talks by Jordan DuBeau and Robert Green
12/11: talks by Michael Wheeler and Lucas Gagnon

Handouts

Group actions, Sylow's Theorem [pdf]

Assignments

due 09/04 [pdf]
due 09/11 [pdf]
due 09/18 [pdf]
due 09/25 [pdf]
due 10/02 [pdf]
due 10/09 [pdf]
due 10/16 [pdf]
due 10/23 [pdf]
due 10/30 [pdf]
due 11/08 [pdf]
due 11/13 [pdf]
due 11/20 [pdf]
due 12/06 [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.