Here is a list of good textbooks that you can consult for reference. I will be posting summaries of my lecture notes on this page. I will not cover the topics in the same order as they are presented in the main textbook, and we will learn some topics that are not covered in the book. The main textbook has an excellent collection of problems and examples, and I encourage you to work through them and ask me or Yueshi if you have difficulty solving any of them.

• Dummit and Foote, Abstract Algebra third edition. (The main text book).
• Isaacs, Algebra .
• Hungerford, Algebra .
• Rotman, An introduction to the theory of groups. (For the group theory part of the course).

The main topics for the fall quarter will be group theory and ring theory (Chapters 1-9 of the textbook). We will also discuss some topics that are not covered in the main book. I know that all of you have taken undergraduate algebra courses before, so this will be a fast-paced course, and we will discuss more advanced topics.

Problem sets will be posted here. Make sure to refresh your bowser. I might change some of the problem sets.

• Due Oct. 3: Here is the problem set.
• Due Oct 10: Here is the problem set.
• Due Oct 17: Here is the problem set.
• Due Oct 24: Here is the problem set.
• Due Oct 31: Here is the problem set.
• Due Nov 7: Here is the problem set.
• Due Nov 14: Here is the problem set.
• Due Nov 21: Here is the problem set.
• Due Dec 2: Here is the problem set.

You can use my lecture notes from 2018.

I will summarize what gets covered during the lectures here. You should go over the notes and your book on the relevant topics.

• 9/24 (1) Lecture 1. Group of symmetries. Symmetric group. Group of automorphisms of a graph. Group of automorphisms of the \(n\)-cycle: dihedral group. Conjugation. Group of automorphisms of a group. Inner automorphisms. Order of a power of an element.

• 9/29 (2) Lecture 2. Group action induced by a homomorphism. Group homomorphism induced by a group action. Left translation action of \(G\) on \(G/H\). Conjugation action of \(G\) on \(G\). Given an action of \(G\) on \(X\), obtaining the left translation action of \(G\) on the space of functions on \(X\). Kernel of a group action. The normal core of a subgroup. Subgroups whose index is the smallest prime factor of the order of a group are normal. Kernel of the Conjugation action is the center. \(G\)-orbits.

• 10/1 (3) Lecture 3. \(G\)-similarity relation. Equivalence classes form a partition. Quotient spaces. The orbit-stabilizer theorem. Generalized class equation. Class equation. Conjugacy classes and centralizers. Conjuagtes of a subgroup and normalizer subgroups. Category of \(G\)-sets and \(G\)-equivariant maps. \(G\)-equivariant map from \(X\) to the set of subgroups of \(G\). \(G\)-equivariant map from \(G\) to the power set of \(X\). The Cauchy-Frobenius theorem on the cardinality of quotient space.

• 10/6 (4) Lecture 4. Assuming the order of \(P\) is a power of a prime \(p\) and \(P\) acts on a finite set \(X\), \(|X|\equiv |X^P| \pmod{p} \). (Cauchy's theorem) Suppose \(p\) is a prime factor of the order of a group \(G\). Then the number of elements of order \(p\) of \(G\) is of the form \(kp-1\); in particular, it is not zero. Suppose \(P\) is a \(p\)-subgroup of a finite group \(G\). Then \([G:P]\equiv [N_G(P):P]\pmod{p}\). In particular, if \(p\) divides \([G:P]\), then \(p\) divides \([N_G(P):P]\), and if \(p\) does not divide \([G:P]\), then \([G:N_G(P)]\equiv 1\pmod{p}\). Next, we proved Sylow's first, second, and third theorem.

• 10/8 (5) Lecture 5. Every group of order \(pq\) where \(p < q\) are primes has a normal subgroup of order \(q\). Definition of a characteristic subgroup. Every characteristic subgroup is normal, but the converse does not hold. Every normal Sylow subgroup is characteristic. If \(P\) is a Sylow subgroup of \(G\) and \(P\subseteq H\) is a subgroup of \(G\), then \(P\) is a Sylow subgroup of \(H\). If \(P_1,P_2\) are Sylow subgroups of \(G\) and \(P_1,P_2\subseteq H\), then there exists \(h\in H\) such that \(P_2=hP_1h^{-1}\). The union of Sylow \(p\)-subgroups of \(G\) is the set of all elements of \(G\) whose order is a power of \(p\). If \(v_p(|G|)=1\), then \(f_p(G)=s_p(G)(p-1)\) where \(f_p(G)\) is the number of elements of order \(p\) of \(G\). For every Sylow \(p\) subgroup of \(G\), we have \({\rm Syl}_p(N_G(P))=\{P\}\). For every automorphism \(\theta\in {\rm Aut}(G)\) and every Sylow \(p\)-subgroup \(P\) of \(G\), \({\rm Syl}_p(\theta(N_G(P)))=\{\theta(P)\}\); in particular, if \(\theta(N_G(P))=N_G(P)\), then \(\theta(P)=P\). For every Sylow \(p\)-subgroup \(P\) of \(G\), \(N_G(N_G(P))=N_G(P)\). Every group of order \(p(p-1)\) has a characteristic subgroup of order \(p\). Every group of order \(p(p+1)\) has a normal subgroup of order either \(p\) or \(p+1\).

• 10/13 (6) Lecture 6. Proved that if a group of order \(p(p+1)\) does not have a normal subgroup of order \(p\), then \(p=2^n-1\) is a Mersenne prime and \(G\) has a normal subgroup of order \(2^n=p+1\). Defined co-chain, co-chain complex, exact co-chain, short exact sequence (SES), standard SES, homomorphism of co-chains, isomorphism of co-chains, and isomorphism of SESs. Proved that every SES is isomorphic to a standard SES. Proved the Short Five Lemma for groups.

• 10/15 (7) Lecture 7. Defined a splitting SES. Showed that a standard SES associated to a normal subgroup \(N\) is equivalent to the existence of a complementary transverse subgroup \(H\) for \(N\). Defined semidirect product of two groups. Described the canonical splitting SES of a semidirect product.

• 10/20 (8)

• 10/22 (9)

• 10/27 (10)

• 10/29 (11)

• 11/3 (12)

• 11/5 (13)

• 11/10 (14)

• 11/12 (15)

• 11/17 (16)

• 11/19 (17)

• 11/24 (18)

• 11/26 (19)

• 12/1 (20)

• 12/3 (21)

• Homework are due on Saturdays 9:00 pm unless it is specified for another time. You should post them in GradeScope.
• Late Homework are not accepted.
• There will be 9 problem sets. Your cumulative homework grade will be based on the best 8 of the 9.
• You may work on the problems with your classmates and discuss them with me or Yueshi, but you must write up your own solutions. Copying from another student's solutions is not acceptable and will be considered cheating. Copying from an online solution bank or using generative AI to produce solutions is not acceptable either. If absolutely necessary, you may use any resource to learn about a solution. However, you must understand the solution and be able to reproduce it on your own. Simply copying a solution is considered cheating.
• Only selected problems will be scored, but you are responsible for understanding all the posted problems (including the ones that are not part of the homework assignments).
• 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.

• Your weighted score is the best of
  • Homework 25%+ midterm 25%+ Final 50%
  • Homework 25%+ Final 75%
• Your letter grade is determined by your weighted score. And it is meant to suggest how your current performance corresponds to your likely result on the qualifying exam to be held next year: A = PhD subject Pass, A- = PhD general Pass, B+= Lower PhD general pass, B= Master’s Pass, B- or less = not likely to pass the qual.

• There is no make-up exam.
• No notes, textbooks, calculators and electronic devices are allowed during exams.
• Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (https://osd.ucsd.edu/.) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments.

• Midterm:
  • Time: Oct 29, 2026.
  • Location: This is an in-class exam. It will take place during the last 50 minutes of class.
  • Topics: All topics discussed in class through the end of the lecture on October 22.
  • Questions will be fairly similar to the homework assignments and the examples discussed in class. Make sure that you know how to solve any of them.
• The final exam:
  • Date: Dec 10,2026, 8:00AM-10:59AM
  • Location: APM 7321.
  • Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in your book.
  • 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.