Math 3330
Section 2.5
Math 3330
Section 2.5
Congruence of Integers
Congruence Modulo n
Let n be a positive integer. For integers x and y, x is congruent to y modulo n if
and only if x y is a multiple of n. We write
x y(mod n )
The integers x and y are con
Math 3330
Section 5.3
The Quotient Field of an Integral Domain
Motivation:
Consider Integers only elements with multiplicative inverses are 1 and -1.
Create fractions rational numbers to be able to divide. But, division is just the inverse of
multiplicati
Math 3330
Section 6.1
Math 3330
Section 6.1
Ideals and Quotient Rings
* Definition Ideal of a Ring*
The subset I of a ring R is called an ideal if it satisfies the following conditions:
1. I is a subring of R
2. For every r R and for every a I , ar I and
Math 3330
Chapter 8
Page 1 of 9
Math 3330
Chapter 8
Polynomials
You all know what a polynomial is, but lets look at polynomials from an abstract algebra
perspective, and discover some of their most important properties.
Before we visit the familiar polyno
Math 3330
Section 4.6
Page 1 of 8
Math 3330
Section 4.6
Quotient Groups
Start with what we have done in the last two days
1. Products of Subsets
Let A and B be nonempty subsets of the group G. Define the product of AB to be the
subset
AB = cfw_x G| x = ab
Math 3330
Section 5.2
Math 3330
Section 5.2
Integral Domains and Fields
* Definition: Integral Domain *
Let D be a ring. Then D is called an integral domain if
1. D is commutative
2. D has a unity e that is not 0
3. D has no zero divisors
Example:
n
when
Math 3330
Section 5.1
Page 1 of 8
Math 3330
Section 5.1
Rings
* Definition of a Ring *
Let R be a nonempty set on which at least one equivalence relation = and two
operations are defined. Call the operations addition + and multiplication *. Then R is
a ri
Math 3330
Section 3.2
Page 1 of 7
Math 3330
Section 3.2
Properties of Group Elements
Reminder: A group is a set with a binary operation which is closed, associative, has an
identity element and in which every element has an inverse.
A group is called an a
3330 Notes Section 3.3
Page 1 of 10
Math 3330
Section 3.3
Subgroups
Definition of a SUBGROUP
Let G be a group with respect to the binary operation *. A subset H of G is called a
subgroup of G if H forms a group with respect to the binary operation * that
Math 3330
Section 3.5
Math 3330
Section 3.5
Isomorphism
Remember the group of permutations on 3 elements?
Compare that group with the group of symmetries on an equilateral triangle.
Page 1 of 8
Math 3330
Section 3.5
Page 2 of 8
Notice the similarities.
Co
Math 3330
Section 3.4
Page 1 of 8
Math 3330
Section 3.4
Cyclic Groups
A group G is cyclic if there exists an element a in G such that
G= a
.
There may be MORE THAN one element in a cyclic group such that G can be written in
this form.
Example Integers
Any
Math 3330
Section 3.6
Page 1 of 5
Math 3330
Section 3.6
Homomorphisms
Isomorphisms are special maps between groups. Are they the only ones?
Why cant we just use any function between the sets?
What special properties does a map need to play nice with the g
Math 3330
Section 4.2
Cayleys Theorem
Theorem: All groups are isomorphic to a group of permutations.
Proof: Let G be a group with operation *.
For an element a of the group G, Consider the map f a
: G G given by
fa (x) = ax
It is a permutation
One-to-one
3330
Section 4.5
Page 1 of 6
Math 3330
Section 4.5
Normal Subgroups
Normal Subgroup
Definition: Let H be a subgroup of G. Then H is a normal subgroup of G if xH=Hx for all
elements x of G.
Note that xH=Hx is SET EQUALITY, not element equality.
Example of
Math 3330
Section 4.4
Cosets of a Subgroup
Part 1 : Products of Subsets Cosets
Let A and B be nonempty subsets of the group G. Define the product of AB to be the
subset
AB = cfw_x G| x = ab , for some a A , b B
Examples:
Properties
(AB)C = A(BC)
AB is not
Math 3330
Section 4.1
Page 1 of 12
Math 3330
Section 4.1
Finite Permutation Groups
Describing a permutation on a set with n elements
A = cfw_a1 , a 2 , a3 , a4 ,., an
How many permutations are there?
Matrix representation of the permutation f:
a1
f (a
UH - Math 3330 - Dr. Heier - Spring 2014
HW 4 - Solutions to Selected Homework Problems
by Angelynn Alvarez
1. (Section 2.4, Problem 3 (d) and (e) Find the great common divisor (a, b) and integers m and n such
that (a, b) = am + bn.
(d) a = 52, b = 124.
S
UH - Math 3330 - Dr. Heier - Spring 2014
HW 3 - Solutions to Selected Homework Problems
by Angelynn Alvarez
1. (Section 1.7, Problem 8) Prove that
xRy if and only if x + 3y is a multiple of 4
is an equivalence relation.
Proof. To prove R is an equivalence
UH - Math 3330 - Dr. Heier - Spring 2014
HW 5 - Solutions to Selected Homework Problems
by Angelynn Alvarez
2. (Section 2.5, Problem 17) Find a solution x Z 0 x < n, for the following congruence.
25x 31(mod7)
Solution. Because gcd(25, 7) = 1, we know ther
UH - Math 3330 - Dr. Heier - Spring 2014
HW 7 - Solutions to Selected Homework Problems
by Angelynn Alvarez
1. (Section 3.3, Problem 14g) Prove that the following subset H of M2 (R) is a subgroup of the group G
of all invertible matrices in M2 (R) under m
UH - Math 3330 - Dr. Heier - Spring 2014
HW 8 - Solutions to Selected Homework Problems
by Angelynn Alvarez
3. (Section 3.5, Problem 26) Prove that any innite cyclic group is isomorphic to Z under addition.
Proof. Let G = a be any innite cyclic group. The
UH - Math 3330 - Dr. Heier - Spring 2014
HW 9 - Solutions to Selected Homework Problems
by Angelynn Alvarez
2. (Section 4.2, Problem 1) Write out the elements of a group of permutations that is isomorphic to G
and exhibit and isomorphism from G to this gr