Syllabus 150B: Modern Algebra
Winter 2006

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Lectures: MWF 10:00-10:50am, Phys/Geo 148   crn: 60710
Discussion section: R (Thurs) 10:00-10:50A, CHEM 176
Instructor: Monica Vazirani, MSB 3224, phone: 752-2218, mjvazirani@ucdavis.edu
OR if it is really really urgent, use my @math address.
Office hours: Wednesday 1:10pm-2:10pm;     Monday 1pm-2pm i also have hours, but may have to give attention to calculus students as well (later in the quarter, this hour may shift to later in the day or to Wednesdays)
in 3224 MSB.
T.A.: Shinpei Baba shinpei@math.ucdavis.edu
Office hours: Tuesday 12:30 - 2pm in MSB 2137
extra office hour March 20, 2-4PM
Eaman Fattouh eamanf@math.ucdavis.edu
Office hours: Mondays from 1:30-2:30pm in 3219 MSB

Text: Michael Artin, Algebra, published by Prentice Hall, 1991. This is on reserve in Shield library.
Problem Sets: There will be weekly homework assignments, generally handed out on Xday, due the following Xday. In class.
You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you do collaborate with students on a problem, please write their names at the end of your homework (collaborators: Alice, Bob, Carol, etc.). No late homeworks will be accepted. Solutions to the problems will be discussed in the discussion section. This is also a good forum to get help with problems and to ask questions!
Exams: Midterm -- February 8, Wednesday.
Final exam: Tuesday, March 21 at 8:00 am Wellman 230 (so they tell me) NOT Phys/Geo 148
There will be no make-up exams!
Grading: The final grade will be based on: Problem sets (and occasional quizzes) 25%, Midterm 35%, Final 40%
Prerequisites: A proof-writing class (such as 108), and a good linear algebra class (such as 22A or 167) , and 150A.
Web: http://www.math.ucdavis.edu/~vazirani/W06/150B.html

Online resources

Peter Scott's group theory tutorial. Each subsection has a quiz at the bottom of the page.

Introduction to permutations, a pdf file, especially the first 10 pages.

Introduction to group theory

Random other .pdf files available: See the last 2 pages of : symmetries of square and hexagon for associating 2 x 2 matrices to the flips and rotations in a nonstandard basis , handout on basis notation like [T]_B^B , a second handout on change of basis , handout on Z/nZ , extra practice problems , (note, R[x]_3 is another notation for P_3(R), polynomials of degree less than or equal to 3) more practice problems ,  
See the last 2 pages of : symmetries of square and hexagon for associating 2 x 2 matrices to the flips and rotations in a nonstandard basis ,
and this Review on the dihedral group  

The handout1 from Jan 4th;
a long (8page) handout on rigid motions including lots of notation, and ways of expressing and composing them. this one was NOT handed out in class so you must print it for yourself.  
A shorter summary of rigid motions and drawing them. this one was NOT handed out in class so you must print it for yourself.
a copy of page 173 with all the wallpaper patterns;
New handouts:
another copy of page 173 with all the wallpaper patterns, annotated with red dots/lines and the point groups marked ; also see now a more detailed explanation (5 pages) of the point groups
Some pages from a book on conjugacy and cycle type in S n . Really only the first 2 pages do this, but the next 3-4 are interesting and have some examples/exercises.

Warning: Artin (and hence Barchechat) multiplies permutations (cycles) right to left instead of left to right. i prefer we multiply them as one would functions. so fg(i) = f( g(i)) = f ° g (i) .

Homework


Homework 1: , due January 13
the following handout might be useful

Solutions by Shinpei: pdf

Homework 2: , due January 20
Solutions by Shinpei: pdf
Homework 3: , due Jan 27
Solutions by Shinpei: pdf

Homework 4: , due Feb 3
Solutions by Shinpei: pdf

Homework 5: , due Feb10
Solutions by Eaman: pdf
The soln to 6.4.1 is here as a study-aid. Soln by Eaman.


Solutions to Quiz 1 are here.

Midterm!!! is Feb 8, in class
Solutions to midterm are here.   [for the wall paper patterns-- the line indicated, you can reflect over and then translate by some amount, so a reflection exists (making it Dn vs Cn), but the line is not the line of the glide reflection. a correct line to flip over is here ...]


Homework 6: , due Feb 17
Solutions by Eaman: pdf

Homework 7: , due Feb 24
Solutions by Eaman: pdf [don't read soln to #9 yet, it needs editing]

Homework 8: , due March 3
Solutions by Shinpei: pdf

Homework 9: , due March 10
Solutions by Shinpei: pdf (updated)


Homework 10: , due March 15

Study groups If you are interested in forming a study/hw group w/ your peers, their contact info is here.
Optional Problems A list of optional problems people have done are here, along, eventually with links to their solutions.

Final Exam: Tuesday, March 21 at 8:00 am Wellman 230 (so they tell me) NOT Phys/Geo 148

And for other practice quizzes, don't forget Peter Scott's group theory tutorial. Each subsection has a quiz at the bottom of the page.

Content of the lectures:

The class is based on Chapters 5-10 of Artin's book. Topics to be discussed include:

5. Symmetry
Symmetry of plane figures
The group of motions of the plane
Finite groups of motions
Discrete groups of motions
Group operations
Operation on cosets
The Counting Formula
Finite Subgroups of the rotation group (Tetrahedral group)

6. Operations of a group on itself
Class equation
Operations on subsets
Sylow Theorems
The Groups of order 12

9. Group Representations (if time permits)
Definition of a group representation
G-invariant subspaces and irreducible representations
Characters
Permutation representations and the regular representation
One-dimensional representations
Schur's Lemma, orthogonality relations

7. Bilinear forms
Definition
Symmetric forms: orthogonality
the Geometry associated to a symmetric form
Hermitian Forms
the Spectral Theorem

8. Linear Groups
The classical linear groups
the special unitary group
orthogonal representation of SU2

10. Rings (if time permits)
Definition
Formal construction of integers and polynomials
Homomorphisms and Ideals
Quotient rings and relations in a ring
Integral domains and fraction fields (if time permits)
Maximal ideals (if time permits)