MATH 463
Group Theory
This is an elective course
on Group Theory. Groups are
introduced in Math 116, and studied
further in Math 367. Students who wants to learn more groups are welcome to
take this course. Students who are planning
for a graduate study in an algebraic field should consider it as a must
course. Students who take this
course usually take the representation theory course Math 464 in the
second semester where they see how linear algebra and group theory get together
in a fruithful way. Groups are used in
other fields, often through group actions. We will introduce this concept early
and use whenever possible.
Frequency: Fall
METU Credit & ECTS Credit: 3 & 6.0
Catalog description: Group, subgroup,
normal subgroup, cyclic subgroup, coset, quotient group. Commutator subgroup,
center, homomorphism and isomorphism theorems (invariant subgroup, wreath
products), Abelian groups. Free abelian group, rank of an abelian group.
Divisible abelian group, periodic Abelian group. Sylow Theorems and their
applications, soluble groups, nilpotent groups.
Course Objectives and outcomes:
This course aims
· to teach the concepts listed in the Catalog Description such as group, group homomorphism, subgroup and quotient group, Isomorphism Theorems, group actions , Sylow Theorems and their applications.
· to improve the proficiency in dealing with abstract concepts, and writing down simple proofs.
· to improve the proficiency in giving examples of the abstract concepts.
By the end of the course the students are expected to be able to:
* understand the statements of the theorems and write the proofs of important theorems when an outline is given, that is, provide the missing arguments.
* write the proofs of important theorems whose proofs are simple .
* give examples and counterexamples illustrating the mathematical concepts presented in the course.
* use basic theorems of group actions in a given setting.
Instructor: Semra Öztürk
Office Hours : See http://users.metu.edu.tr/sozkap/aa.pdf
Grading :
MidTerm1: 25 Points (Oct. 2016 )
MidTerm2: 25 Points (Nov. 25,
17:40, 2016 )
MidTerm3: 25 Points (Dec. 20, 2016 )
Final Exam: 30 Points (Jan., 2017 )
Quiz/Attendance/HW: 20 Points
*Best two midterms will be counted
Suggested
textbooks:
• http://www-groups.mcs.st-andrews.ac.uk/~martyn/teaching.html
• Rose, Harvey E. A course on finite groups (available through METU library http://link.springer.com/book/10.1007%2F978-1-84882-889-6 )
• John S. Rose, A Course on Group Theory (Dover Publications, New York 1994), QA171.R7. (not available in METU library)
• Joseph J. Rotman, The Theory of Groups: An Introduction, Allyn and Bacon, Boston 1965), QA171.R67,
• Derek J. S. Robinson, A Course in the Theory of Groups (Second Edition), Graduate Texts in Mathematics 80 (Springer–Verlag, New York 1996), QA171.R568
• B. A. F. Wehrfritz, Finite Groups: A Second Course on Group Theory (World Scientific, Singapore 1999), QA177.W44,
• M. I. Kargapolov & Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics 62 (Springer–Verlag, New York 1979), QA171.K3713
Useful Links:
http://www-groups.mcs.st-andrews.ac.uk/~martyn/5824/5824lecturenotes.pdf
http://www-groups.mcs.st-andrews.ac.uk/~martyn/5824/5824problemsheets.pdf
http://www-groups.mcs.st-andrews.ac.uk/~martyn/4003/4003lecturenotes.pdf
http://www.math.uconn.edu/~kconrad/blurbs/
http://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Group_2.pdf
http://www.math.niu.edu/~beachy/aaol/groups.html
http://users.metu.edu.tr/matmah/Graduate-Algebra-Solutions/grouptheory-1.pdf
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Development_group_theory.html
Current Semester Course Home Page: http://users.metu.edu.tr/sozkap/463/
|
Week |
Dates |
Syllabus(Math 463)
2016-1 |
|
|
1 |
Oct 03-07 University Opening Ceremony:
10:00 (8:40, 9:40 and 10:40
classes will not be held) |
Revision
and Rectivation :Group, subgroup,
homomorphism, quotient group , normal
group |
|
|
2 |
Oct 10-14 Add-Drop
and Advisor Approvals |
Homomorphisms, Isomorphism
theorems |
|
|
3 |
Oct 17-21 |
Group Actions and
Permutation Representations |
|
|
4 |
Oct 24-28 National Holiday (Republic
Day) (October 28, Friday afternoon holiday) |
Group Actions and
Permutation Representations |
|
|
5 |
Oct 31-Nov 04 |
Sylow Theorems Applications of Sylow Theorems |
|
|
6 |
Nov 07-11 Commemoration of Atatürk (Thursday) |
Sylow Theorems Applications of Sylow Theorems |
|
|
7 |
Nov 14-18 |
Sylow Theorems Applications of Sylow Theorems |
|
|
8 |
Nov 21-25 |
The Jordan-Hölder Theorem |
|
|
9 |
Nov 28-Dec 02 |
Direct Products, Semidirect
Products |
|
|
10 |
Dec 05-09 |
Direct Products, Semidirect
Products |
|
|
11 |
Dec 12-16 |
Direct Products, Semidirect Products |
|
|
12 |
Dec 19-23 |
Soluble
groups |
|
|
13 |
Dec 26-30 New Year’s Day (Sunday) |
Soluble groups , Nilpotent
groups |
|
|
14 |
Jan 02-06 |
Nilpotent groups |
|
|
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