Instructor 
Dragos Oprea

Lectures: 
WF 5:00  6:20 PM, PETER 1D03 
Course Assistants 
Jasper Bird
 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.

Course Content 
First course in a rigorous threequarter 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.

Prerequisities:  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. 
Grade Breakdown  The grade is computed as the
best out of the following
weighed average:
 Homework 20%, Midterm I 20%, Midterm II
20%,
Final Exam 40%.
 Homework 20%, Best Midterm 20%,
Final Exam 60%.

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


Midterm Exams  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.

Final Exam  The final examination will be held on
Thursday, December 12, 710 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  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. 
Announcements and
Dates 
 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.

Exams 
 Preparation for Midterm 1:
 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 Summaries  
Lecture 1: Wellordering 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 ngon. 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.
