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This is the first semester algebra
course here at Tech. The goal of this course is to introduce students
to groups, rings and fields. These are fundamental objects that have a
beautiful and subtle structure of their own and are the "right"
abstract language to express may other things that you will
encounter in the rest of your mathematical studies (like certain spaces
of functions, symmetries, counts of subspaces...). It is assumed that
students have already had an undergraduate abstract algebra course. So
while we will cover everything "from the beginning", the basic material
will be covered very rapidly.
Announcements:
- Please take part in the Course/Instructor Opinion Survey (CIOS).
Written comments are most appreciated and will help shape future
versions of this course (and others). Please comment on specific things
you liked and did not like about the course. Your feedback is most
appreciated!
- The final exam will be Thursday, December 14 from
8:00 to 10:50 am in Skiles 271. The final will be comprehensive and
look more or less like a qualifying exam, but the topics covered since
the last test will be emphasized.
- The second midterm will be in class on November 9.
- The first midterm will be in class on September 26.
Lectures:
Tuesday
and Thursday 9:35 in Skiles 271.
Professor:
John Etnyre
Office: Skiles 163
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu
Grading for the class will be based on
- weekly homework
assignments (20%),
- two midterm exams (25% each) and
- a final exam (30%).
The assignments will be posted on Tuesdays and
will be due the following Tuesday in class. I encourage you
to work on these assignments with other students in the class and to
use whatever other resources you might have, but each problem must be written up
in your own words by you. At some point everyone
needs to learn TeX or LaTeX so I
encourage you to write up your homework using one of these packages,
but this is not a requirement. If you would like help getting started
with TeX or LaTeX you are welcome to talk to me about it.
The dates for the exams will be announced later.
The textbook for this class is
"Abstract Algebra" (third edition) by Dummit and Foote. If you have an
older edition of the book that is probably OK, the main thing is to
check with me or other students in the class to make sure you are
working the correct homework problems.
Homework
Assignment 1
(due August 29):
- Section 1.1: 9, 22, 25, 36
- Section 1.2: 7, 10
- Section 1.3: 1, 11, 13
- Section 1.4: 10
- Section 1.6: 9, 20, 24
- Using notation from problem 20 above prove:
Aut(Z)=Z/2Z, Aut (Z/6Z)=Z/2Z, Aut(Z/8Z)=(Z/2Z) X (Z/2Z) and
Aut(Z/pZ)=Z/(p-1)Z for p prime. Here = means isomorphic.
- Section 2.1: 14, 15
Homework
Assignment 2 (due September 5):
- Section 2.2: 7
- Section 2.4: 6, 7
- Section 3.1: 1, 3, 5, 14, 21, 36
- Prove there are exacty two groups of order 4.
What are they?
- Prove that any group of order 2n (n some
positive
integer) has an element of order 2. If the group is abelian and n is
odd, then show that this element is unique.
Homework
Assignment 3 (due September 12):
- Section 2.2: 6, 12
- Section 2.4: 14, 15
- Section 3.2: 4, 5
- Section 3.3: 7
- Section 4.1: 1, 6
- Section 4.2: 2, 9
Homework
Assignment 4 (due September 19):
- Section 4.2: 8
- Section 4.3: 5, 8, 13, 23, 24
- Section 4.5: 7, 12, 16
- Show that groups of oder 12, 28, 56, 200 are
not simple.
- How many elements of order 7 are there in a
simple group of order 168?
Homework
Assignment 5 (due September 26):
- Study for first midterm.
- Section 3.5: 12, 15
- Section 4.4: 3, 5, 13
- Section 4.5: 31, 34
Homework Assignment 6 (due October 10):
- Section 4.6: 2, 3
- Section 5.1: 14, 15, 18
- Section 5.2: 2a, 2b, 3a, 3b, 4a, 4b
- Section 5.4: 11
- Section 5.5: 6, 8
Homework Assignment 7 (due October 19):
- Section 5.5: 9
- Section 6.1: 3, 7, 21, 24, 25
- Section 6.3: 5, 14
Homework Assignment 8 (due October 24):
- Section 7.1: 7, 8, 15, 23, 26, 27
Homework Assignment 9 (due October 31):
- Sectoin 7.1: 13, 14
- Sectrion 7.2: 2, 3, 9, 10
- Section 7.3: 1, 2, 6, 12
Homework Assignment 10 (due November 7):
- Section 7.3: 17, 29, 30
- Section 7.4: 7, 8, 13, 15, 30
Homework Assignment 11 (due November 14):
- Section 7.4: 31, 37, 40
- Section 7.5: 4
- Section 8.1: 3, 6, 7, 10
Homework Assignment 12 (due November 21):
- Section 8.2: 3, 4, 6
- Section 8.3: 6, 11
- Section 9.1: 5, 13
- Section 9.2: 3, 5
Homework Assignment 13 (never due):
- Section 9.3: 1, 4
- Section 9.4: 12, 13, 20
- Section 13.1: 2, 5
- Section 13.2: 2, 4, 7, 10, 14, 20
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