Math 103A: Algebra I

Summer 2021

 Lectures: Will be pre-recorded and posted in YouTube. Part of the scheduled lecture time will be used as office hour and taking exams.

Office hours:
 M 9:30-10:50 W 9:30-10, 11-12 Meeting ID: Password: 941 4349 8051 8 factorial
TA's information:
 Name E-mail adddress Meeting ID Cameron Cinel ccinel ucsd edu TBA
Cameron's office hours: TBA
TA's information:
 Name E-mail adddress Meeting ID Abhik Pal apal ucsd edu TBA
Abhik's office hours: TBA

• Title: Abstract Algebra I: Introduction to Group Theory.
• Credit Hours: 4.
General information

• Prerequisite: Math 109 or Math 31CH. Summer courses are extremely fast pace. So enroll only if you can devote the necessary time slot.
• Catalog Description: First course in a two-quarter introduction to abstract algebra. In this course, we study basics of group theory: subgroups, homomorphisms, factor groups, first isomorphism, etc.
Book

• John B. Fraleigh, A First Course in Abstract Algebra.
Calendar

This is only a tentative schedule and it might change during the quarter.

Lectures

• Introduction, equivalence relation, partition. PDF
• Congruences, division algorithm. PDF
• GCD,Euclid's algorithm, Euclid's lemma. PDF
• Multiplicative structure of $$\mathbb{Z}_n$$. PDF
• What is a group? Basic properties of groups. PDF
• Groups and symmetries. PDF
• Homomorphisms and subgroups. PDF
• Cyclic groups, order of elements, and subgroups of cyclic groups. PDF
• Group isomorphism. PDF
• Symmetric groups: cycle decomposition, support of permutations, the linking relation, conjugation. PDF
• Parity of permutations. PDF
• Further properties of order of elements and order of permutations. PDF
• Group actions, orbits, conjugacy classes, cosets, Lagrange's theorem. PDF
• Normal subgroups,factor groups, and the first isomorphism theorem. PDF
• Automorphism group. PDF
• Finite abelian groups. PDF
• Fixed points of fintie groups of order $$p^n$$. Cauchy's theorem, and Sylow's first theorem. PDF
Lectures and Topics by date

• M 6/28 Juneteenth.
• W 6/30:
• Introduction and equivalent relations.
• Here is the link to the lecture.
• Here is the lecture note.
• Congruences, well-ordering principle, long division, addition and multiplication on the set of integers modulo $$n$$.
• Here is the link to the lecture.
• Here is the lecture note.
• Greatest Common Divisors, Euclid's algorithm, and Euclid's lemma.
• Here is the link to the lecture.
• Here is the lecture note.
• M 7/5: Independence day.
• W 7/7:
• Quiz 1.
• Multiplicative structure of $$\mathbb{Z}_n$$.
• Here is the link to the lecture.
• Here is the lecture note.
• What is a group?
• Here is the link to the lecture.
• Here is the lecture note.
• Groups and symmetries.
• Here is the link to the lecture.
• Here is the lecture note.
• M 7/12:
• Group homomorphisms and subgroups.
• Here is the link to the lecture.
• Here is the lecture note.
• W 7/14:
• Quiz 2.
• Cyclic groups, order of elements, and subgroups of cyclic groups.
• Here is the link to the lecture.
• Here is the lecture note.
• Group isomorphisms. Finite cyclic groups up to an isomorphism. Examples. Cayley's theorem.
• Here is the link to the lecture.
• Here is the lecture note.
• Computation in Symmetric Groups.
• Here is the link to the lecture.
• Here is the lecture note.
• M 7/19:
• Parity of permutations.
• Here is the link to the lecture.
• Here is the lecture note.
• Order of permutations.
• Here is the link to the lecture.
• Here is the lecture note.
• W 7/21:
• Quiz 3.
• Group actions, orbits, conjugacy classes, cosets, Lagrange's theorem.
• Here is the link to the lecture.
• Here is the lecture note.
• Normal subgroup and factor groups.
• Here is the link to the lecture.
• Here is the lecture note.
• M 7/26:
• Factor groups and the first isomorphism theorem.
• Here is the link to the lecture.
• Here is the lecture note.
• Automorphism group.
• Here is the link to the lecture.
• Here is the lecture note.
• The standard form of a fintie abelian group.
• Here is the link to the lecture.
• Here is the lecture note.
• W 7/28:
• Cauchy's and Sylow's theorem.
• Here is the link to the lecture.
• Here is the lecture note.
• F 7/30: Quiz 4.

Homework

• Homework will be assigned in the assignment section of this page.
• Homework are due on Fridays at 9:00 pm, through GradeScope .
• Late homework is not accepted.
• There will be 4 problem sets. Your cumulative homework grade will be based on the best 3 of the 4.
• Selected problems on the each assignment will be graded.
• Style:
• A messy and disorganized homework might get no points.
• You can scan or simply take a clear photo of your homework and upload it
• You must select pages corresponding to your solutions of problems during the upload process.
• 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. You can use Overleaf . Overleaf is an easy to use and an excellent online LaTeX editor.
• 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.
quizzes.

• There will be 4 quizzes throughout the session.
• You will write them on Wednesdays 10-10:50 or 19-19:50, except the last exam which will be on Friday 10-10:50 or 19-19:50. Before the first quiz, you have to let me know if you will be taking the quizzes at 12 or 19.
• No collaboration with other humans or with online resources is allowed.
• No (e-)notes, textbooks, and calculators are allowed during exams.
• 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.
• Solutions should be clearly written and you must select pages corresponding to your solutions of problems during the upload process. For the quizzes, your solutions should be hand-written.
• You will access the quizzes in GradeScope. Students will be divided in two main groups, and will be assigned different exams accordingly.
• Students who take the earlier quiz are not allowed to share their exams or discuss the problems with others till the second group is done with the quiz.
• You should use the same Zoom link as the one provided for the office hours. Either your TAs or I will be on the Zoom meeting. Your cameras should be on for the duration of quizzes. The cameras should show your workplace. Students with the AFA will be assigned to separate breakout rooms.

• The first quiz:
• Date: 7/7
• Topics: All the topics covered till the end of 6/30 lectures.
• Here is the first version of Quiz 1, and Here is its solutions.
• Here is the second version of Quiz 1, and Here is its solutions.
• Here are the scores: 30 (5 students), 29, 28 (7 students), 27 (2 students), 26, 25 (4 students), 24 (2 students), 23 (3 students), 22 (5 students), 21 (2 students), 20 (2 students), 19 (2 students), 18.5, 18 (2 students), 17 (2 students), 16 (3 students), 15 (2 students), 13 (2 students), 12 (2 students), 8 (2 students), 1.
• The median is 22 and the average is 21.42
• Special thanks is due to those 13 students who got 28 and above.
• For this exam, above 25 can be considered in the A+,A,A- range, above 18 can be considered in the B+,B,B- range, and below 10 is extremely alarming. ( Of course, these are not your final grades. I am mentioning them only to give a feeling on how you have done in the first quiz.)
• The second quiz:
• Date: 7/14
• Topics: All the topics covered till the end of 7/12 lectures.
• Here is the first version of Quiz 2, and Here is its solutions.
• Here is the second version of Quiz 2, and Here is its solutions.
• Here are the scores: Total Score 29 (4 students), 28, 27, 26.5, 25.5, 24, 22, 21, 20.5, 20, 19.5, 19, 18 (2 students), 17.5, 17 (3 students), 16.5, 16 (3 students), 15.5, 15, 14.5, 14 (4 students), 13 (4 students), 12, 11 (2 students), 10.5, 10 (3 students), 9, 8 (2 students), 6 (2 students), 5 (2 students), 0
• The median is 15.50 and the average is 15.93
• Special thanks is due to those 8 students who got 25.5 and above.
• For this exam, 18 and above can be considered in the A+,A,A- range, 12 and above can be considered in the B+,B,B- range, and below 8 is extremely alarming. ( Of course, these are not your final grades. I am mentioning them only to give a feeling on how you have done in the second quiz.)
• The third quiz:
• Date: 7/21
• Topics: All the topics covered till the end of 7/19 lecture.
• Here is the first version of Quiz 3, and Here is its solutions.
• Here is the second version of Quiz 3, and Here is its solutions.
• Here are the scores: Total Score 29.5, 27, 25, 25,23.5, 23, 23, 22.5, 22.25, 21.5, 21, 19.5, 19, 18, 18, 18, 17, 16, 16, 15.5, 15, 15, 15, 14.5, 14, 14, 13.5, 13.5, 13.5, 13, 12.5, 12, 11, 10, 10, 10, 10, 9.75, 9.5, 9.5, 9, 9, 8, 7.5, 6, 3, 2.5.
• The median is 14.5 and the average is 15.12.
• Special thanks is due to those 4 students who got 25 and above.
• For this exam, 18 and above can be considered in the A+,A,A- range, 11 and above can be considered in the B+,B,B- range, and below 8 is extremely alarming. ( Of course, these are not your final grades. I am mentioning them only to give a feeling on how you have done in the second quiz.)
• The fourth quiz:
• Date: 7/30
• Topics: All the topics that are discussed in the course.

• There will be 4 quizzes. Your cumulative quiz grade will be based on the best 3 of the 4.
• Your final weighted score is
• Homework 20%+ Quiz grade 80%
• 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.
• If you wish to have your homework or quizzes regraded, you must request regarde through Gradescope within the specified window of time. No regrade will be accepted after the deadline.
• Submitting a regrade request without a legitimate explanation may result in the loss of one point on the given problem.
Further information
• There is no make-up exam.
• No notes, textbooks, calculators and electronic devices are allowed during exams.
• Academic Integrity: 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.
Assignments