MATH 411 Algebra-I

The class meets Tu and Th at 11:15 in LGRT 204.

Professor: Jenia Tevelev

Office: LGRT 1236. Office hours are M 3-5:30

E-mail: tevelev(at)math.umass.edu

Textbook: A first course in abstract algebra by John B. Fraleigh, 7th edition. We will cover the material related to groups (Chapters I, II, III, and VII). Rings and fields will be studied in Math 412.

It is very important to understand everything written in the textbook. Read slowly, take your time understanding all definitions, examples, and proofs. Don't take anything on faith. Each time you read a statement of a theorem or lemma, pause and think about it. Do you know definitions of all concepts mentioned in the statement? Why is it non-trivial? Have you seen similar statements? How would you prove it? When you read a proof, try to isolate its main idea. Remember that each problem is unique but there aren't so many different proof techniques. A trick used in the proof of some theorem can be later used in the proof of a homework problem.

The HW will be posted on the class website. It will be collected each week on Tuesday. No late homework will be accepted. The grader will select five random problems in the homework and grade them. So it is important to solve all problems. All solutions in the homework and on the exams should be written in complete English sentences. Each time you use a special proof technique such as proof by induction or proof by contradiction, say so. If you want to use a known theorem or lemma in your argument, simply quote its number from the textbook.

It is important to take lecture notes of all classes. I will mostly follow the textbook, however quite often I will give a different example, or explain an alternative approach.

There will be two midterms, on Oct 12 (Tuesday) and Nov 15 (Monday), from 6pm to 8pm, room TBA.

The final grade will be 25% midterm 1, 25% midterm 2, 25% final exam, 20% HW, 5% class participation. Two worst HW grades will be dropped. I routinely ask students to do problems at the blackboard. When it is your turn to do the blackboard, do not try to solve the problem completely in your head before writing it down. Instead, try to generate as many ideas as possible and brainstorm the problem in collaboration with other students.

HW assignments:

HW due on	Sections to read	Homework problems	Remarks
September 14	Section 0 (Sets and Relations)	4,6,10, 11,12, 13,14,15, 16,17, 24,25, 29,32,33, 35,36 (from Section 0)
September 21	Section 1 (Introduction and Examples), Section 2 (Binary Operations), Section 3 (Isomorphic Binary Structures)	9, 18, 32, 36, 37, 38, 40 (Section 1); 2, 10, 12, 20, 24, 26, 28, 34 (Section 2); 2, 7, 8, 14, 18, 26, 32 (Section 3)
September 28	Section 4 (Groups), Section 5 (Subgroups)	6, 8, 10, 14, 24, 25, 28, 31, 32, 34, 36 (Section 4); 4, 8, 10, 20, 22, 26, 32, 34, 39, 40, 42, 50, 51 (Section 5)
October 5	Section 6 (Cyclic Groups)	2, 11, 14, 16, 18, 20, 22, 28, 30, 32, 34, 36, 40, 44, 46, 48, 50, 52, 56a (Section 6)	The first midterm will be on October 12, 6-8pm, in LGRC A201 It will cover sections 1-7.
October 12	No HW due - prepare for the midterm! The practice exam is available here.
October 19	No HW due. During the class, I will ask three people to solve the three most difficult problems from the midterm. It is available here.
October 26	Section 8 (Groups of Permutations), Section 9 (Orbits, Cycles, and the Alternating Groups)	4, 8, 10, 18, 21, 24, 26, 35, 40, 41, 46, 47, 49 (Section 8); 2, 6, 8, 10, 13, 16, 23, 27ab, 29, 34, 35, 39 (Section 9)
Nov 2	Section 10 (Cosets and the Theorem of Lagrange), Section 11 (Direct Products and Finitely Generated Abelian Groups)	6, 7, 9, 10, 16, 19, 20, 24, 26, 28, 29, 30, 34, 37, 38, 40 (Section 10); 6, 10, 14, 18, 24, 32, 36, 46, 50, 52, 53, 54 (Section 11)
Nov 9	Section 12 (Plane Isometries), Section 13 (Homomorphisms)	6, 8, 16, 17, 26, 30, 34, 38, 40 (Section 12); 2, 4, 5, 8, 10, 12, 18, 22, 24, 32, 36, 44, 47, 48, 50, 53 (Section 13)
Nov 16	No HW due - prepare for the midterm! The exam will be on Monday, November 15, from 6pm to 8pm, in LGRC A201. It will cover sections 8-13. We will have a problem session during the class on November 10. The practice exam is available here. No solutions will be posted, but you are welcome to discuss these problems with me (and get hints) during the office hours. I will have office hours on Friday, November 12, from 12:30 to 2:30. There will be no office hours on the day of the exam. On Tuesday November 16 (the day after the midterm), I will ask three people to solve the three most difficult problems from the midterm at the blackboard. The midterm is available here.
Nov 23	Section 14 (Factor Groups)	From section 14: 6, 10, 14, 16, 21, 22, 23, 24, 26, 30, 31, 33, 34, 36, 38
Nov 30	Section 15 (Calculations with Factor Groups)	From section 15: 4, 10, 12, 14, 19, 24, 25, 30, 34, 36, 37, 38, 39
Dec 7 (last homework)	Sections 16 (Group Actions), 17 (Applications to Counting)	From section 16: 1, 2, 3, 8, 6, 10, 13, 15, 18; From section 17: 2, 3, 4, 5, 8, 9