Math 100A: Algebra I

Fall 2019

 Lectures: T-Th 3:30 PM--4:50 PM APM B402 A Office hours: T-Th 4:50 PM--6:00 PM APM 7230

 Discussion session information:

 A01 Tu 6:00 PM--6:50 PM APM 7321 Jacob Naranjo janaranj ucsd edu A02 Tu 7:00 PM--7:50 PM APM 7321 Jacob Naranjo janaranj ucsd edu

 TA's office hour information:

 Jacob Naranjo; Tue 12-2, Wed 1-3 APM 6414

General information

• Title: Abstract Algebra I: Introduction to Group Theory.
• Credit Hours: 4.
• Prerequisite: Math 109 or Math 31CH. Math 20F is also important as linear transformations provide us a rich set of examples. Math 100 is a difficult and time consuming course, so enroll only if your course load allows it.
• Catalog Description: First course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. In this course, we study basics of group theory: subgroups and factor groups, homomorphisms, isomorphism theorems, groups acting on sets, and others.
Book

• Keith Nicholson, Introduction to abstract algebra (4th edition).
• There are lots of interesting books on group theory. I like the following books:
• P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic abstract algebra. (It has nice problem sets.)
• Joseph Rotman, An introduction to the Theory of Groups. (It is an advanced book, full of interesting topics. Towards the end, some of the connections of group theory and topology is explored.)
Schedule

This is a tentative schedule for the course. If necessary, it may change.

• Lecture 01: Overview of what a group is. Review of division algorithm and divisibility. Subgroups of $$\mathbb{Z}$$. (Section 1.2)
• Lecture 02: Review of the greatest common divisor of two integers. Euclid's lemma. Fundamental theorem of arithmetic (Section 1.2).
• Lecture 03: There are infinitely many primes. Number of divisors of $$n$$, $$\sqrt{2}$$ is not a rational number. Number of positive divisors of $$n$$ is odd if and only if $$n$$ is a perfect square. Congruence arithmetic. Remainder of division by $$9$$. (Section 1.2).
• Lecture 04: Remainder of division by 9 or 11. Equivalence relations and equivalence classes. Residue classes mod $$n$$. $$(\mathbb{Z}_n,+,\cdot)$$. Units in $$\mathbb{Z}_n$$ (Sections 0.4, 1.3).
• Lecture 05: $$\mathbb{Z}_n^{\times}$$. Group homomorphism. Direct product of finitely many groups. Chinese Remainder Theorem. Kernel of a group homomorphism. Wilson's theorem. Fermat's little theorem. (Section 1.3).
• Lecture 06: Computation in the symmetric group. The set of points that are moved by a permutation. Cycles. Linking relation. Disjoint permutations commute. (Section 1.4).
• Lecture 07: Cycle decomposition. Transposition. Parity. (Section 1.4)
• Lecture 08: More examples of groups: $$S^1$$, $$\mu_n$$. Exponent laws, cancellation laws. General subgroup criterion. Isomorphism. Conjugation. Automorphisms and inner automorphisms. Computation in groups. (Section 2.2)
• Lecture 09: Catch up (important subgroups: image, kernel) (Section 2.3). Exam 1.
• Lecture 10: Centralizer of an element. Intersection of subgroups. Center of a group. $$Z(S_n)=\{1\}$$. Center of $${\rm GL}_n(\mathbb{R})$$. Center of abelian groups. Subgroup generated by one element. Order of an element. Order of an element in a finite abelian group. (Sections 2.3, 2.4)
• Lecture 11: Properties of order of an element. Order of $$g^m$$. Generators of a cyclic group. $$C_n\simeq \mathbb{Z}_n$$. (Section 2.4)
• Lecture 12: Order of product of two commuting elements. Order of a permutation. Subgroup structure of a cyclic group. $$\mathbb{R}^+\simeq \mathbb{R}$$, $$\mathbb{R}\setminus \{0\}\not \simeq \mathbb{R}$$(Section 2.4)
• Lecture 13: Cayley's theorem. Group of automorphisms. Inner automorphisms. Cosets. (Section 2.5)
• Lecture 14: Index of a subgroup. Lagrange's theorem. Some of the consequences of Lagrange's theorem. Euler's theorem. Normal subgroup. (Section 2.6)
• Lecture 15: Tower of subgroups. Normal subgroups. Kernel is normal. Subgroups of index 2 are normal. Center is normal. (Sections 2.6, 2.8)
• Lecture 16: Product set. Normal subgroup with trivial intersection. Chinese Remainder Theorem (2nd version). (Section 2.8).
• Lecture 17: Factor group. Exam 2. (Section 2.9)
• Lecture 18: The first isomorphism theorem. $$G/Z(G)\simeq {\rm Inn}(G)$$. The second isomorphism theorem. The third isomorphism theorem. (Sections 2.10, 8.1)
• Lecture 19: The third isomorphism theorem. The correspondence theorem. (Section 8.1)
• Lecture 20: Group actions. Space of orbits is a partition. The orbit-Stabilizer Theorem. Conjugacy classes. The class equation. Center of a non-trivial finite $$p$$-group is non-trivial. (Sections 8.2, 8.3)
Homework

• Homework will be assigned in the assignment section of this page.
• Homework are due on Thursdays at 5:00 pm. You should drop your homework assignment in the homework drop-box in the basement of the APM building
• Late homework is not accepted.
• There will be 8 problem sets. Your cumulative homework grade will be based on the best 7 of the 8.
• Selected problems on the each assignment will be graded.
• Style:
• A messy and disorganized homework might get no points.
• The upper right corner of each assignment must include:
1. Your name (last name first).
2. Your discussion session (e.g. A01,etc.).
3. Homework assignment number.
• Full-sized notebook papers should be used.
• All pages should be stapled together.
• Problems should be written in the same order as the assignment list. Omitted problems should still appear in the correct order.
• As a math major, sooner or later you have to learn how to use LaTex. I really encourage you to use Latex to type your solutions.
• A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.
• You can work on the problems with your classmates, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating.
• Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture.
• Homework will be returned in the discussion sections.

• Your weighted score is the best of
• Homework 20%+ midterm exam I 20%+ midterm exam II 20%+ Final 40%
• Homework 20%+ The best of midterm exams 20%+ Final 60%
• You must pass the final examination in order to pass the course.
• Your letter grade is determined by your weighted score using the best of the following methods:
•  A+ A A- B+ B B- C+ C C- 97 93 90 87 83 80 77 73 70
•  Based on a curve where the median corresponds to the cut-off B-/C+.
• If more than 90% of the students fill out the CAPE questioner at the end of the quarter, all the students get one additional point towards their weighted score.
• Homework and midterm exams will be returned in the discussion sections.
• If you wish to have your homework or exam regraded, you must return it immediately to your TA.
• Regrade requests will not be considered once the homework or exam leaves the room.
• If you do not retrieve your homework or exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.
Further information
• There is no make-up exam.
• Keep all of your returned homework and exams. If there is any mistake in the recording of your scores, you will need the original assignment in order for us to make a change.
• No notes, textbooks, calculators and electronic devices are allowed during exams.
• Academic Dishonesty: Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university. It is in your best interest to maintain your academic integrity.
Exams.

• The first exam:
• Time: October 24, in class exam.
• Topics: Sections 1.2, 1.3, 1.4, and whatever is covered in class that are related to these sections.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Extra office hour: Wednesday 9a-10a.
• Practice: besides going through your homework assignments, examples presented in the class and problems in the relevant chapters of your book, you can use the following practice exams:
• From Professor Rogalski, Fall 2015, PDF .
• From Professor McKernan, Fall 2016, PDF .
• The practice exam and the first midterm from Professor Oprea are more relevant for our course as we are using the same text book:
• The second exam:
• Time: November 21, in class exam.
• Topics: Till the end of the basic properties of normal subgroups.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Extra office hour: Wednesday 9a-10a.
• Practice: go through your homework assignments, examples presented in the class and problems in the relevant chapters of your book. You can find practice exams here:
• From Professor Rogalski, Fall 2015, PDF .
• From Professor McKernan, Fall 2016, PDF .
• The practice exam and the second midterm from Professor Oprea are more relevant for our course as we are using the same text book:
• The final exam:
• Time: December 09, 3:00pm-5:59pm.
• Location: APM B402 A.
• It is your responsibility to ensure that you do not have a schedule conflict involving the final examination. You should not enroll in this class if you cannot sit for the final examination at its scheduled time.
• Topics: All the topics that are discussed in the class and in the book. Except group actions the rest of topics are part of the final exam.
• Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them.
• Extra office hour: Friday 10-11a.
• Practice: besides going through your homework assignments, examples presented in the class and problems in the relevant chapters of your book, you can find practice exams here:
• From Professor Rogalski, Fall 2015, PDF .
• From Professor McKernan, Fall 2016, PDF .
• As it is stated above, in the worst case scenario the median of the weighted scores corresponds to the B-/C+ cut-off.
Lectures
Brian Chao has kindly provided his lecture notes to share with you. He mentioned that "anyone who found typos or have suggestions is welcome to email b7chao at ucsd dot edu". Here is the link to his lecture notes.
Assignments
• Homework 1 (Due October 3)
• Section 1.1: 2(d), 5, 10(c).
• Section 1.2: 7(a), 9(a), 11, 12, 19, 26, 28, 33.

• Homework 2 (Due October 10)
• Section 1.2: 22, 35, 39, 43, 45.
• Section 1.3: 1(f), 2(b), 4, 21(a), 21(b), 33.

• Homework 3 (Due October 17)
• Section 1.3: 9(a), 13, 22(b), 25(a), 27(a), 28, 32, 34(a), 34(e).

• Homework 4 (Due October 31)
• Section 1.4: 1(f), 4, 13(a), 13(f), 18, 21(c), 26, 27(c), 27(d), 30.

• Homework 5 (Due November 7)
• Section 2.2, 1(c), 1(d), 7, 10(b), 11(a), 22, 25.
• Section 2.3, 1(i), 24, 25.

• Homework 6 (Due November 20) (I made an exception and extended this week's homework deadline.)
• Section 2.3, 8, 13.
• Section 2.4, 2(b), 4(b), 7(b), 8(b), 20(b), 23(a,b), 25.

• Homework 7 (Due November 26)
• Section 2.5, 4, 6, 8(a), 23, 25, 28.
• Section 2.6, 10(b), 35(a), 36.
• (Bonus Problem) Suppose $$G$$ is a finite group and for any positive integer $$n$$, there are at most $$n$$ elements of $$G$$ such that $$g^n=1$$. Prove that $$G$$ is cyclic.

• Homework 8 (Due December 5)
• Section 2.8, 9, 11, 12, 13(a), 25(b), 25(c), 26(a).
• Section 2.9, 7, 11, 12.
• Section 2.10, 21, 22.
• Show that $${\rm Inn}(G)$$ is a normal subgroup of $${\rm Aut}(G)$$.
• Show that if $$G/Z(G)$$ is cyclic, then $$G$$ is abelian.