Math 100A

• Office: APM 6-101
• Office hours: Tuesday 1 - 2pm, Wednesday 10 - 11am. You can also speak to me after lecture.
• Email: doprea at math dot ucsd dot edu

• Discussion: APM 2402, Wednesday, 7:00 - 7:50 PM and Wednesday, 8:00 - 8:50 PM.
• Office: APM 2230
• Office hour: Wednesday 11 - 2, 3 - 4.
• Email: j1bird at ucsd dot edu.

First course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include groups, subgroups, factor groups, homomorphisms, isomorphism theorems, groups acting on sets, and others.

Math 31CH or Math 109 or permission of instructor. Students will not receive credit for both Math 100A and Math 103A. Math 100 is a difficult and time consuming course, so enroll only if your course load allows it.

• Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.

• Homework 20%, Best Midterm 20%, Final Exam 60%.

Introduction to Abstract Algebra by W. Keith Nicholson, 4th Edition.

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. It will be expected that you read the assigned material in advance of each lecture.

Homework problems will be assigned on the course homework page. There will be weekly problem sets, typically due on Friday at 4:30 PM in the TA's dropbox in the basement of the APM building. The due date of some of the problem sets may change during the quarter depending on the pace at which we cover the relevant topics. One problem set will be dropped in the calculation of the homework grade.

You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.

Style: please make sure you write your solutions neatly, write your name, and staple all pages togehter.

It is very important to do the problem sets in order to do well on the exams. Selected problems on the each assignment will be graded.

There will be two midterm exams given on October 18 and November 15. There will be no makeup exams. No notes, textbooks, calculators and electronic devices are allowed during exams.

Regrading policy: graded exams and homeworks will be handed back in section. If you wish to have your homework or exam regraded, you must return it immediately to your TA. Regrading is not possible after the exam leaves the room.

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.

The final examination will be held on Thursday, December 12, 7-10 PM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.

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.

• Friday, Sept 27: First lecture
• Friday, October 18: Midterm I
• Monday, Nov 11: Veterans' Day. No class.
• Friday, November 15: Midterm II
• Thursday - Friday, Nov 28 - 29: Thanksgiving. No class.
• Friday, December 6: Last Lecture
• Thursday, December 12, 7pm - 10 pm: FINAL EXAM.

• Preparation for Midterm 1:
• Practice Problems for Midterm 1 - PDF
• Midterm I and Solutions

• Preparation for Midterm 2:
• Practice midterm from Fall 2015 (from Professor Rogalski) - PDF
• Practice midterm from Fall 2016 (from Professor McKernan) - PDF
• Practice Problems for Midterm 2 - PDF
• Midterm II and Solutions

• Preparation for Final Exam:
• Practice Final from Fall 2015 (from Professor Rogalski) - PDF
• Practice Final from Fall 2016 (from Professor McKernan) - PDF
• Additional practice Problems for the Final - PDF
• Final and Solutions

• Lecture 1: Well-ordering principle. Divisibility of integers. Division with remainder. Greatest common divisor.
• Lecture 2: Greatest common divisor exists. Greatest common divisor as a linear combination. Coprime integers. Prime numbers. Euclid's lemma. The fundamental theorem of arithmetic.
• Lecture 3: There are infinitely many primes. The sieve of Eratosthenes. Equivalence relations and equivalence classes. Congruences mod n. Addition and multiplication in Z_n.
• Lecture 4: Invertible elements in Z_n. Finding inverses using Euclid's algorithm. All nonzero elements in Z_p are invertible. Wilson's theorem. Chinese Remainder Theorem. Fermat's little theorem. When is -1 a quadratic residue mod p.
• Lecture 5: Permutations. Symmetric group. Composition of permutations. Cycles. Shape of permutations using directed graphs.
• Lecture 6: Decomposition of permuations into disjoint cycles. Transpositions. Every permutation is product of transpositions. Partity theorem. Odd/even permutations and some properties.
• Lecture 7: Proof of the parity theorem. Binary laws. Groups. Examples. First properties of groups.
• Lecture 8: More examples of groups: symmetric group, Klein group, cyclic group, product of groups. Subgroups and examples. Subgroup tests.
• Lecture 9: Lattice of subgroups. Intersection of subgroups. Center. Conjugate subgroups. Normal subgroups. Cyclic subgroups. Order of elements. Examples.
• Lecture 10: Properties of the order. The subgroup generated by an element has size equal to the order. Order of powers of elements. Order of permutations.
• Lecture 11: Classification of subgroups of the cyclic group. Homomorphisms and isomorphisms. Examples.
• Lecture 12: Reinterpretation of the Chinese Remainder Theorem. Basic properties of homomorphisms and isomorphisms. Kernel and image. Examples.
• Lecture 13: More on isomorphisms. Injectivity of homomorphisms via the kernel. Automorphisms form a group. Examples: automorphisms of cyclic groups (infinite and finite). Euler's function. Inner automorphisms.
• Lecture 14: Left and right cosets. Index of subgroups. Lagrange's theorem. Orders of elements divide the order of the group. Groups of prime order are cyclic.
• Lecture 15: The dihedral group. Symmetries of the regular n-gon. Classification of groups with 2p elements for p prime.
• Lecture 16: Normal sugroups and examples: center, kernels, subgroups of index 2. Quaternion group. Simple groups. Abelian simple groups. Criterion for a group to be product of two of its normal subgroups. Abelian groups with p^2 elements.
• Lecture 17: Normal subgroups and quotient groups. The natural homomorphism from the group to the quotient. Examples. Criterion for abelian groups via cyclicity of quotients by subgroups of the center.
• Lecture 18: The three isomorphism theorems and their proofs.