Week

Material

Homework

has to be uploaded on BB not later then Thursday 1:15 PM of the corresponding week Only the RED problems will be graded

August 24th -- August 28th

Chapter 1. Introduction to Groups, p. 16

August 31st -- September 4th

Chapter 2. Subgroups, p. 46

·       Problem Set 1
Section 1.1: 9, 22.
Section 1.2: 4, 10.
Section 1.3: 2, 13.
Section 1.4: 2, 11.
Section 1.6: 9 (Hint: Use Exercise 1.2.4 and 1.3.13 assigned above.), 23.
Section 1.7: 8, 10.

·September 7th - September 11th

Chapter 3. Quotient Groups and Homomorphisms, p. 73

       Problem Set 2
Section 2.1: 6, 10.
Section 2.2: 4, 14.
Section 2.3: 21, 26.
Section 2.4: 7, 12.
Section 2.5: 11, 16.

September 14th -- September 18th

Special week because of the quiz

DIAGNOSTIC QUIZ in Thursday lecture.

Chapter 4. Group actions, p. 112

·       Problem Set 3
Section 3.1: 17, 24.
Section 3.2: 4, 12.
Section 3.3: 3, 8.
Section 3.4: 7, 8.
Section 3.5: 3, 16.

September 21st -- September 25th

Chapter 5. Direct and Semidirect Products and Abelian Groups, p. 152

·       Problem Set 4
Section 3.4: 9, 10.
Section 3.5: 6, 8.
Section 4.1: 3, 7(a) and (d).
Section 4.2: 11, 12, 13.

September 28th -- October 2nd

Chapter 11. Vector Spaces, p. 408
MIDTERM 1 in Thursday lecture.
No assigned problems to be collected this week.

October 5th -- October 9th

Chapter 11. Vector Spaces (continued), p. 408

·       Problem Set 5
Section 4.3: 19, 21, 24.
Section 4.4: 2, 10.
Section 4.5: 24, 38.
Section 4.6: 3, 4.

October 12th -- October 16th

Inner Product Spaces and The Spectral Theorem

·       Problem Set 6
Section 4.6: 6, 7.
Section 5.1: 7, 8*, 9*.
Section 5.2: 8, 14.
Section 5.4: 8, 9.
* There is a small mistake in Problems 8,9 of Section 5.1. Please prove injectivity of the homomorphism from Problem 8 only for the special case occurring in Problem 9.

October 19th -- October 23rd

Chapter 7. Introduction to Rings, p. 223

·       Problem Set 7
Section 5.5: 7, 13, 23.
Section 11.1: 6, 8, 9.
Section 11.2: 19, 26, 33.

October 26th -- October 30th

Chapter 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains, p. 270

·       Problem Set 8
Section 11.3: 2, 3, 4.
Section 11.4: 4, 5.
Section 11.5: 5, 11, 13.

November 2nd – November 6th

Chapter 9. Polynomial Rings, p. 295

·       Problem Set 9
Problem Set on the spectral theorem ( pdf)

November 9th -- November 13th

Chapter 15. Commutative Rings and Algebraic Geometry, p. 656

MIDTERM 2 in Thursday lecture.
No assigned problems to be collected this week.

, November 16th -- November 20th

Semisimple Rings and Wedderburn's Theorem
Chapter 10. Introduction to Module Theory, p. 337

·       Problem Set 10
Main Problem. Let G and G' be groups with the same finite order n. Let m be some integer divisor of n, and assume that G and G' each contain a unique normal subgroup P, resp. P', of order m. For each group, consider the associated group which is the centralizer of P, resp. P', modulo the center of P, resp. the center of P'. If G and G' are isomorphic, prove that these associated groups are also isomorphic.

Next, assume that there exists a subgroup Q of G which intersects P in only the identity element and such that P and Q generate G, i.e., G is a semidirect product of P and Q. Let f denote the induced homomorphism from Q to the outer automorphism group of P (not the usual homomorphism to the automorphism group of P). Show that the kernel of f is canonically isomorphic to the quotient of the centralizer of P by the center of P. Thus the isomorphism class of the kernel of f is an isomorphism invariant of G. In particular, observe that the hypotheses hold if we restrict to groups G and G' which are each a semidirect product of a fixed finite p-group P and a fixed group Q whose order is less than p+1. Moreover, if P is abelian then the outer automorphism group of P equals the usual automorphism group of P so that f is the usual homomorphism.

Second Problem. For every problem on Midterm 2 where you lost points, please write up a complete, correct solution to that problem (if you only lost points on a part of the problem, you can write up just the solution for that part). If you got full credit on the exam, you do not need to write up anything (you will automatically get credit for this part of the homework assignment).

Chapter 7.1 23,24,25,26

Chapter 7.2 8,9,10,1

Chapter 7.3 29,30,33,36

Chapter 7.4 31,32,33,41

Chapter 7.5 2,6

Chapter7.6 6,8,9,10,11

Chapter8.1 5,7,8,12

November 23th -- November 27^th

Thanksgiving Break: No classes in session.

November 30th -- December 4th

Chapter 12. Modules over Principal Ideal Domains, p. 456

·       Problem Set 11
This problem set is to be assigned. Section 12.1: 16, 17, 18, 19 (Please read through this sequence of exercises.)
Section 12.3: 2, 9, 13, 14, 17.