Math 8202: Class Outlines

o 03/03/10: (Your regular instructor Professor Bobbe Cooper takes over from now on.) Sylvester's Theorem, signature, quadratic forms. [Kostrikin & Manin: part of Sections 2.3-4; Lang: Sections XV.2, XV.4-5]

o 03/01/10: Homework collected. Spectral theorems for symmetric and self-adjoint operators. The orthogonal and unitary groups, orthogonal bases for pairs of symmetric (Hermitian) forms. [Kostrikin & Manin: part of Sections 2.7-8; Lang: Sections XV.3, XV.6-7 (XIV.12-13 in 2nd edition)]

o 02/26/10: Transpose, adjoint, symmetric, and self-adjoint, or Hermitian linear maps, their eigenvalues. [Kostrikin & Manin: part of Sections 2.3 and 2.8; Lang: Sections XIII.5-7 and XV.1, part of XV.6-7 (XIV.12-13 in 2nd edition)]

o 02/24/10: Bilinear Forms: symmetric, skew-symmetric, alternating, Hermitian, kernel, rank, nondegenerate. [Kostrikin & Manin: Sections 2.2 and 2.4; Lang: Sections XIII.6-7 and XV.1 (XIV.1 in 2nd edition)]

o 02/22/10: Homework collected. Quiz. Bilinear Forms: matrices, base change. [Kostrikin & Manin: Section 2.2.; Lang: Section XIII.6]

o 02/19/10: Normal subgroups in Sn for n > 4. The simplicity of alternating groups. The nonsolvability of symmetric and alternating groups. [Section 4.6]

o 02/17/10: The uniqueness of nonabelian groups of order pq. Another application of Sylow's Theorems: classification of groups of order 30. [Parts of Sections 4.5 and 5.5]

o 02/15/10: Homework collected. Basics of direct and semidirect products. Applications of Sylow's Theorems: Cauchy's Theorem, groups of order pq. [Sections 5.1, 5.4; parts of Sections 5.5 and 4.5]

o 02/12/10: Proof of Sylow's Theorems I and III. [Section 4.5 through p. 141. Note that the proof I presented in class was quite different from that in the textbook. Choose the one to your liking.:-)]

o 02/10/10: Sylow's Theorems. Proof of Sylow's Theorem II. [Part of Section 4.5: Definitions and the wording of Theorem 18, Lemma 19, proof of (2). Note that the proof of (2) in class largely overlaps with the proof in the text, but is different.]

o 02/08/10: Homework collected. Quiz. Aut (Zn). [Section 4.4 through Proposition 17(1)]

o 02/05/10: Automorphism groups. Inner automorphisms. Inn (Sn) and Aut (Sn). [Section 4.4 through the statement of Proposition 16]

o 02/03/10: Problem 3.4.11. [Solvable Groups and the Derived Series through Proposition 10 from Section 6.1]

o 02/01/10: Homework collected. The Class Equation. Groups of order p2. [Section 4.3, excluding Simplicity of A5 and Right Group Actions]

o 01/29/10: Conjugation action of a group on itself and its subgroup lattice. Conjugacy classes, conjugate elements and subgroups, centralizer, normalizer. Conjugacy is Sn. Counting conjugacy classes in S4. [Section 4.3, excluding the Class Equation, Simplicity of A5, and Right Group Actions]

o 01/27/10: Cycle decomposition. Left regular action. [Sections 4.1-2]

o 01/25/10: Group actions. [Section 4.1 before Cycle Decomposition]

o 01/22/10: The Hölder Program. The extension problem. Solvable groups. Transpositions. The alternating group. [Sections 3.4-5]

o 01/20/10: Introduction. Cauchy's Theorem for abelian groups. Simple groups. The Feit-Thompson theorem. Composition series. The Jordan-Hölder theorem. [Dummit and Foote: Section 3.4 through p. 103]


Last modified: Wed Jan 20 16:21:24 CDT 2010