A representation of a group G is a group action of G on a vector space V by invertible linear
maps. For example, the group of two elements Z_2=cfw_0,1 has a representation phi by phi(0)v=v
and phi(1)v=-v. A representation is a group homomorphism phi:G->GL
MAT1100
Algebra I
Assignment 1
Contents
1.
2.
3.
4.
Problem 1
Question 2
Question 3
Question 4
Tyler Holden - Fall 2011
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1. Problem 1
If g is an element of a group G, the order |g| of g is the least positive number n such
that g n = 1. If x, y G
MAT1100 Homework 1
Ling-Sang Tse
Due Date: October 6, 2014
Problem 1
Case 1: |xy| = n < for some n N
Claim: |yx| = n.
Proof:
(yx)n+1 = y(xy)n x = yx
Multiplying both sides by (y)1 on the left and x1 on the right,
(yx)n = e.
Since the order n of an element
MAT1100
Algebra I
Assignment 2
Contents
1. Problem 1
1.1. Part a
1.2. Part b
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5
6. Problem 6
7. Problem 7
7.1. Part a
7.2. Part b
8. Source Code
Tyler Holden - Fall 2011
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1. Problem 1
MAT1100
Algebra I
Assignment 4
Contents
1. Problem
2. Problem
2.1. Part a
2.2. Part b
3. Problem
4. Problem
5. Problem
6. Problem
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Tyler Holden - Fall 2011
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Tyler Holden
Mat1100, Assignment 3
1. Problem 1
Clearly all principal id
MAT1100
Algebra I
Assignment 3
Contents
1. Problem
2. Problem
2.1. Part a
2.2. Part b
3. Problem
4. Problem
5. Problem
5.1. Part a
6. Problem
6.1. Part a
6.2. Part b
7. Problem
7.1. Part a
7.2. Part b
8. Problem
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Tyler Holden - Fall 2011
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Homework 2
MAT1100
Due: October 16, 2014
Ling-Sang Tse
Solution to Problem 1
a) n = 11.
Let = (a11 a12 .a1n1 ).(ak1 .aknk ), where is decomposed into disjoint cycles, k is the
number of disjoint cycles, and ni is the length of each disjoint cycle.
The ord
Problem 2. Let G be a group and let Z(G) denote its center.
1. Show that if G/Z(G) is cyclic then G is Abelian.
Let xZ(G) be the generator for Z(G). Since Z(G) G (obvious) we know
from understanding cosets that any g G can be written as xm z for some
xm G
A module is a mathematical object in which things can be added together commutatively by
multiplying coefficients and in which most of the rules of manipulating vectors hold. A module is
abstractly very similar to a vector space, although in modules, coef
Solution to Problem 1
Notation: Let (a,b) be the notation used to denote the ideal generated by a and b.
1. Suppose I is principal i.e., (3, x3 x2 + 2x 1) = (f (x), where f(x) = a0 + a1 x + . + an xn ,
ai s Z. Since 3 (f (x),
3 = (a0 + a1 x + . + an xn )(
A group G is a finite or infinite set of elements together with a binary operation (called
the group operation) that together satisfy the four fundamental properties of closure,
associativity, the identity property, and the inverse property. The operation
If there exists a one-to-one correspondence between two subgroups and subfields such that
G(E(G^')
=
G^'
=
E^',
(1)
E(G(E^')
(2)
then E is said to have a Galois theory.
A Galois correspondence can also be defined for more general categories.
A category co
Solve the following questions
Problem 1. (easy)
Prove that in any ring, (-1)^2=1.
Prove that even in a ring without a unit, (-a)^2=a^2.
(Feel free to do the second part first and then to substitute a=1).
Problem 2.
(Qualifying exam, April 2009) Prove that
Do not turn this page until instructed.
Math 1100 Core Algebra I
Term Test
University of Toronto, October 25, 2011
Solve the 4 problems on the other side of this page.
Each problem is worth 25 points.
You have an hour and fty minutes to write this test.
N