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Fall 2003

Readings

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Text

Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. ISBN: 9780130047632.

The exercises listed below point to the text and are a suggestion from the professor to the students in every given lecture. These exercises were not turned in and were not corrected.

Course readings and exercises.
SES #	TOPICS	READINGS	EXERCISES
1	Groups, matrices, permutations	Chapter 2, section 1; chapter 6, section 6 up to 6.10	Chapters 1, 1.7, 2.12; chapters 2, 1.1b, 1.5, 2.19
2	Subgroups, cyclic groups	Chapter 1, section 4; chapter 2, section 2	Chapters 1, 4.5; chapters 2, 2.1, 2.3, 2.7; chapters 6, 6.9
3	Isomorphisms	Chapter 2, sections 2, 3	Chapters 2, 3.1, 3.7, 3.10, 3.12; chapters 6, 6.2, 6.4
4	Homomorphisms	Chapter 2, sections 4, 5	Chapters 2, 4.1, 4.3, 4,6, 4.16, 5.6, 5.7
5	Cosets	Chapter 2, section 6	Chapters 2, 5.3, 6.7, 6.12, 7.7
6	Quotient groups	Chapter 2, sections 8, 10	Chapters 2, 6.4, 8.3, 10.8, 10.6
7	Modular arithmetic, fields	Chapter 2, section 9; chapter 3, section 2	Chapters 2, 9.2, 9.5; chapters 3, 2.1, 2.3
8	Fields	Chapter 2, section 9; chapter 3, section 2	Chapters 2, 9.8; chapters 3, 2.8, 2.14, 2.17
9	Vector spaces, bases, dimension	Chapter 3, sections 1, 3	Chapters 3, 3.2, 3.4, 3.5, 3.8, 3.11
10	Computation with bases	Chapter 3, section 4	Chapters 3, 4.1–4.3, 4.12
11	The dimension formula	Chapter 4, sections 1, 2	Chapters 4, 1.1, 1.2, 1.4, 2.1, 2.2, 2.4, 2.5
12	First quiz
13	Linear operators, Eigen vectors	Chapter 4, section 3	Chapters 4, 3.1, 3.3–3.5, 3.8
14	The characteristic polynomial	Chapter 4, sections 4, 6	Chapters 4, 4.1, 4.3, 4.4, 6.1, 6.5c, d, 6.7
15	Rotations	Chapter 4, section 5	Chapters 4, 5.1–5.3, 5.5, 5.8
16	Isometries	Chapter 4, section 5; chapter 5, sections 1, 2	Chapters 4, 5.13; chapters 5, 1.3a, c, 2.1, 2.2, 2.9
17	Isometries of the plane	Chapter 5, sections 1, 2	Chapters 5, 1.1a, c, 2.1, 2.2, 2.4, 2.7, 2.9
18	Finite groups of isometries	Chapter 5, section 3	Chapters 5, 3.1–3.3, 3.6, 4.1
19	Discrete groups	Chapter 5, section 4	Chapters 5, 4.2–4.4, 4.6, 4.11
20	Discrete groups (cont.)	Chapter 5, section 4	Chapters 5, 4.8–4.10, 4.12, 4.13
21	Group operations	Chapter 5, sections 5, 6, 7	Chapters 5, 5.3, 5.8, 6.1, 6.5, 7.1–7.3
22	Finite rotation groups	Chapter 5, section 9	Chapters 5, 9.2–9.5, 9.6a, 9.7
23	The class equation	Chapter 6, section 1	Chapters 6, 1.1, 1.2, 1.4–1.6
24	Permutation representations	Chapter 5, section 8	Chapters 5, 8.1, 8.3, 8.5, 9.6b
25	The alternating group	Chapter 6, sections 2, 6	Chapters 6, 2.1, 2.3, 2.5, 6.3, 6.13
26	Second quiz
27	Bilinear forms	Chapter 7, sections 1, 2	Chapters 7, 1.2–1.5, 2.1, 2.2, 2.7
28	Symmetric forms	Chapter 7, section 3	Chapters 7, 2.9, 2.10, 2.21, 3.2, 3.11
29	Hermitian forms	Chapter 7, section 4	Chapters 7, 4.2, 4.5, 5.8, 4.10, 4.15, 4.18
30	The spectral theorem	Chapter 7, section 5	Chapters 7, 5.2, 5.4, 5.5a, 5.7, 5.16
31	Quadrics	Chapter 7, section 6	Chapters 7, 6.1, 6.2, 6.4
32	The special unitary group SU₂	Chapter 8, sections 1, 2	Chapters 8, 1.1, 1.5, 2.1–2.3, 2.5
33	The rotation group	Chapter 8, section 3	Chapters 8, 3.1, 3.4, 3.5, 3.10
34	One-parameter groups	Chapter 4, section 8; chapter 8, section 5	Chapters 8, 5.2–5.4, 5.6
35	One-parameter groups (cont.)	Chapter 4, section 8	Chapters 8, 5.5, 5.7, 5.9
36	Third quiz
37	The lie algebra	Chapter 8, section 6	Chapters 8, 6.1, 6.2, 6.5–6.7
38	Dimensions of the linear groups	Chapter 8, section 7	Chapters 8, 7.1a, b, d, 7.2, 7.10
39	Simple groups	Chapter 8, section 8	Chapters 8, 8.2a, 8.3, 8.5