Math 541
Algebra
Spring 2013
Instructor: Shamgar Gurevich, 317 VV.
Time and Location: Lectures: VAN VLECK B135, 09:30 AM - 10:45 AM, TR
O ce Hours:
Tuesday 11am
1pm
Texts: I will use the book "Algebra" by M. Artin. Mainly chapters 2, 5, 6, 9, 10.
Syllabus
Assignment 3 Final version Parts 1, 2, & 3 Math 541
Due Friday, Sept. 28, at the beginning of class
The following
Section 1.6:
Section 2.4:
Section 1.3:
Section 1.4:
problems from the textbook:
1, 2, 3, 4, 5, 7, 8
15
5, 6, 7, 8, 9
1, 2, 3, 4, 7, 8, 16, 21
Assignment 1 Final version Parts 1, 2, 3, & 4 Math 541
Due Friday, Sept. 14, at the beginning of class
(1) Consider the symmetric group on 3 things, S3 .
(a) How many elements does it have?
(b) Label these as you wish and write out a multiplication table
Assignment 2 Final version Parts 1, 2, & 3 Math 541
Due Friday, Sept. 21, at the beginning of class
The following problems from the textbook:
Section 1.5: 1, 3, 4, 6, 10, 13. For question 13(b), explain why the statement is incorrect
and prove the correct
Assignment 6 Parts 1 & 2 Math 541
The following problems from the textbook:
Section 2.4: 5, 6, 9, 11
Section 2.5: 12, 15, 19, 21
(1) Show that every A O2 (R) has determinant 1. (Hint: look at det(ATA).
(2) The goal of this exercise is to determine the cen
Assignment 6 Part 1 Math 541
(1) Show that every A O2 (R) has determinant 1. (Hint: look at det(ATA).
(2) The goal of this exercise is to determine the centre of Dn (n 3). It will be useful
to use the fact that every element of Dn is of the form i rj with
Assignment 10 Parts 1, 2, & 3 Math 541
Due Friday, Nov. 30, at the beginning of class
The following problems from the textbook:
Section 4.3: 4, 5, 6, 15 and 18, 20, 21
(1) Suppose : R R is a homomorphism of rings. Show that ker() R and
im() R . (This was
Assignment 11 Parts 1 & 2
The following problems from the textbook:
Section 4.5: 8, 10, 12
(1) This will show up in class on Monday. Let : R R be a ring homomorphism
and let I be an ideal in R. Give an example where (I ) is not an ideal in R . But
show th
Assignment 9 Parts 1, 2, & 3 Math 541
Due Monday, Nov. 26, at the beginning of class
The following problems from the textbook:
Section 4.1: 14, 15, 16
Section 4.2: 1, 3, 81
(1) The goal of this problem is to classify groups of order 8.
(a) There are three
Assignment 8 Final version Parts 1, 2, & 3 Math 541
Due Monday, Nov. 12, at the beginning of class
(1) Recall from assignment 6 question (4) that, for h1 , h2 G, we say that h2 is conjugate to h1 if there is g G such that
h2 = g 1 h1 g.
(The map from G to
Assignment 7 Parts 1, 2, & 3 Math 541
Due Friday, Nov. 2, at the beginning of class
The following
Section 2.4:
Section 2.6:
Section 2.7:
problems from the textbook:
13, 14, 18, 24, 31 and 29
7, 8, 11, 12, 13 and 1, 3, 4
2, 3
(1) Let Q be the quaternion gr
Assignment 4 Final version Parts 1, 2, & 3 Math 541
Due Friday, Oct. 5, at the beginning of class
The following problems from the textbook:
Section 2.1: 1, 8, 9, 16
Section 2.3: 1, 2, 3, 8 and 5, 6, 7, 12, 13, 14, 16
(1) Consider the four matrices 1, i, j
Assignment 5 Final version Parts 1, 2, & 3 Math 541
Due Wednesday, Oct. 17, right before the midterm
The following problems from the textbook:
Section 2.5: 1 and 6, 14, 28 and 26
(1) Consider the group C whose elements are the non-zero complex numbers and
Math 541 Spring 2013
Homework#1 Answers, 02/14/13
Denition. A group is a triple (G; ; 1G ); where G is a set, : G G ! G; (g; h) 7! g h; is a
map, called operation, and 1G 2 G a specic element, called identity, such that the following
axioms are satised:
A
Math 541 Spring 2013
Solutions to HW2 Rotational Symmetries
Feb 25, 2013
1. Rotational symmetries.
(a) Consider the collection R of all matrices of the form
r=
cos( )
sin( )
sin( )
,
cos( )
2 R:
10
) with denotes multiplications of matrices,
01
is a group
Math 541 Spring 2013
Preparation for the Mid-Term Test
Remarks.
Answer ALL the questions (a) and (b) below, and only one of the (c)
s
s
s.
Denition (subsections (a) is just a denition and there is no need to justify it. So
just write it down.
Answers to s
Math 541 Spring 2013
HW3 Answers Rotational Symmetries, Subgroups of GL2(R)
03/04/2013
Consider the group (R; ; I ) of rotations of the plane R2 of all matrices of the form
r=
with
cos( )
sin( )
sin( )
,
cos( )
denotes multiplications of matrices, and I =
Math 541 Spring 2013
Preparation for the Final Test
Remarks
Answer all the questions below.
A denition is just a denition there is no need to justify it. Just write it down.
Unless it a denition, answers should be written in the following format:
s
Write
Math 541 Spring 2013
Homework#4 Solutions
03/16/13
1. The orthogonal group. Denote by h ; i the standard inner product on V = R2 : Consider
the set O = fA 2 M at(2; R); such that h Au; Av i = h u; v i for every u; v 2 R2 g; i.e.,
the set of all 2 2 real m
Math 541 Spring 2013
Solutions HW8 Orbits,Cosets, Lagrange Theorem, Fermat
s
s
Little Theorem
04/29/13
Remark. Answers should be written in the following format:
0) Statement and/or Result.
i) Main points that will appear in your explanation or proof or c
Math 541 Spring 2013
Solutions
HW7, Normal Subgroups, Lagrange Theorem, Groups
s
of Prime Order
April 29 2013
Remark. Answers should be written in the following format:
0) Statement and/or Result.
i) Main points that will appear in your explanation or pro
Math 541 Spring 2013
HW#5 Solutions Orthogonal Symmetries of the n-Regular
Polygon, Subgroups of Z; Product, Cyclic Groups
4/6/13
Remark. Answers should be written in the following format:
i) Statement and/or Result.
ii) Main points that will appear in yo
Math 541 Spring 2013
Solutions HW6 - 04-12-2013
1. Homomorphism.
(a) Let : G ! H;
composition =
: H ! K be two homomorphisms of groups. Show that the
: G ! K is also homorphism.
1. Main points. Denition.
def
2. Proof. We have (g g 0 ) = (
)(g g 0 ) =
is h
Math 541 Spring 2013
HW9
Solutions Counting, Quotient Groups
May 13, 2013
1. Counting. Recall that if ' : G ! G0 is a homomorphism the
(1)
#G = # ker(') # Im('):
Consider the nite eld with prime p elements Fp = f0; 1; :; p
modulo p:
1g; with + and
(a) Den
Assignment 12 Parts 1 & 2 Math 541
Due Friday, Dec. 14, at the beginning of class
The following problems from the textbook:
Section 3.2: 2, 3, 9
Section 3.3: 1, 5
(1) In this question, R is a commutative ring with identity. This question continues the
ide