Download LECTURE NOTES FOR INTRODUCTION TO ABSTRACT ALGEBRA

LECTURE NOTES FOR INTRODUCTION TO ABSTRACT ALGEBRA,
SMA205
DAMIAN M MAINGI
1. COURSE DESCRIPTION
DURATION: 45 hours
LECTURER: Damian Maingi
Indivative Content:
Binary operations, Elementary properties of integers, congruences, congruence modulo,
Introduction to algebraic structures Semigroup, monoid, groups and subgroups.
Prerequisite: Basic Mathematics-SMA101 and Plane and Analytic Geometry-SMA105.
Background: When students join university for their undergraduate degree, the algebra they know goes back to when they were solving for unknowns/indeterminates and
specifically in high school when they were solving quadratic equations. Now we take a
turn by defining binary operations on sets with a view to introducing algebraic structures
like groups, fields and vector spaces which will recur in other algebra courses. Analytic
Geometry allows representation of points in a plane by pairs of numbers and points of a
space by triples. Also numbers represent or determine lines, conics in the plane as coefficients. In this course we define operations on sets in measurement/coordinization.
OUTLINE:
1: Preliminary Sets, Mappings, Binary Operations, Algebraic laws, Relations
2: Properties of Integers Postulates of integers, Divisibility, Prime factors, Greatest
Common Divisor, Least Common Multiple, Congruences of integers.
3: Some Algebraic Structures Monoids, Semigroups, Groups
ASSESMENT: Cat 30, Exam 70marks
REFERENCE MATERIAL:
Norman Bloch: Abstract Algebra with Applications
I N Herstein: Topics in Algebra
Date: May 26th, 2014.
Key words and phrases. Groups, subgroups, group homomorphisms and isomorphisms.
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DAMIAN M MAINGI
2. CHAPTER ONE
Notation 2.1. (I) We shall use ∗ to represent a binary operation on a set S.
(II) S represents a set and most of the sets that we shall use are;
(a) N, the set of natural numbers
(b) Z, the set of integers
(c) Q, the set of rational numbers
(d) R, the set of real numbers
(e) C, the set of complex numbers
(f) S + is set of positive elements of S, a number set.
2.1. Preliminaries.
Definition 2.2. A set S is a colletion of items (elements,objects) with a common property.
Definition 2.3. An ordered pair (x, y) of elements is such that (x, y) 6= (y, x) and (x1 , y1 ) =
(x2 , y2 ) if and only if x1 = x2 and y1 = y2 .
Definition 2.4. For 2 non-empty sets X and Y , the cartesian product X × Y is the set
of all ordered pairs (x, y) of element x ∈ X and y ∈ Y i.e. X × Y = {(x, y)/x ∈ X, y ∈ Y }
Definition 2.5. A subset f of X × Y is a mapping from X to Y if and only if for each
x ∈ X there exists a unique y ∈ Y such that (x, y) ∈ f and usually we write y = f (x).
Definition 2.6. A binary operation ∗ on a set S is a well defined function on the cartesian
product of S onto itself whose image is in S
∗
/ x∗y ∈S
i.e. (x, y) ∈ S × S
In other words, a binary operation on a set is a function that acts on 2 elements of the set
to produces a 3rd element which is unique in the set.
Remark 2.7. A binary operation ∗ on a set S satifies 2 axioms i.e.
• it is well defined or x ∗ y is possible for all x, y ∈ S
• closure holds i.e. x ∗ y ∈ S.
Examples of binary operations
(a) Addition, subtraction and multiplication on S, where S = Z, Q, R are binary operations.
(b) Define ∗ on N by x ∗ y = x − 2y then closure does not hold.
LECTURE NOTES FOR INTRODUCTION TO ABSTRACT ALGEBRA, SMA205
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(c) Define ∗ on Q by ab ∗ dc = a+c
then ∗ is not a binary operation for it is not unique
b+d
1
2
1
i.e. 2 ∗ 3 = 5 which is not equal to 24 ∗ 13 = 37
(d) ∗ defined on the set of integers by x ∗ y = xy is not a binary operation since both
closure and uniqueness do not hold.
(e) ∗ defined on the set of rationals by x ∗ y =
not defined if x = y = 0
x
y
is not a binary operation since ∗ is
2.2. Algebraic Laws.
Definition 2.8. Let ∗ be a binary operation on a set S we have the following definitions;
(1) Closure: x ∗ y ∈ S for all x, y ∈ S
(2) Associative: x ∗ (y ∗ z) = (x ∗ y) ∗ z for all x, y, z ∈ S
(3) Commutative: x ∗ y = y ∗ x for all x, y ∈ S
(4) Identity: e ∈ S is said to be the identity if e ∗ a = a ∗ e = a for all a ∈ S
(5) Inverse: a ∈ S is said to be the inverse of b ∈ S if a ∗ b = b ∗ a = e
(6) A binary operation ? is said to be distributive over ∗ if x ? (y ∗ z) = (x ? y) ∗ (x ? z)
for all x, y, z ∈ S
Examples
(1) Define ∗ on Q+ by x ∗ y = xy
. Test for above laws.
5
Solution:
(i) Since Q is closed with respect to multiplication then xy
= x ∗ y ∈ S thus closure
5
holds.
1
(ii) Next associativity, we have x ∗ (y ∗ z) = x ∗ ( yz
) = 15 x(yz)
= (xy)z
5
5
5 5
= (x ∗ y) ∗ z since multiplication is associative
yx
(iii) x ∗ y = xy
=
=
y
∗
x
5
5
(iv) Let e be the identity, then for any a ∈ Q, a = e ∗ a = ea
, imlpying e = 5.
5
(v) Now suppose a ∗ b = e = 5 i.e.
ab
= 5 this implies that b = 25
.
5
a
+
hQ , ∗i is called an abelian or commutative group.
(2) Define ∗ on Z by x ∗ y = x + y − 1.
Do closure, associative, commutative, identity and inverse laws hold in Z w.r.t ∗?
Solution:
(i) Closure holds since Z is closed w.r.t + and - so x + y − 1 = x ∗ y ∈ Z.
(ii) For associative law we have, x ∗ (y ∗ z) = x + (y ∗ z) − 1 = x + (y + z − 1) − 1
= (x + y − 1) + z − 1 = (x ∗ y) + z − 1 = (x ∗ y) ∗ z
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DAMIAN M MAINGI
(iii) x ∗ y = x + y − 1 = y + x − 1 = y ∗ x, i.e. commutative laws holds.
(iv) Let e ∈ Z be the identity, then ∀ a ∈ Z we have a = a ∗ e = a + e − 1 i.e. e = 1.
(v) Inverses: Suppose a ∗ b = 1 i.e. a + b − 1 = 1 then b = 2 − a is the inverse of a.
(3) Let ∗ be a binary operation on Z defined by x ∗ y = xy + 5 ∀, x, y ∈ Z.
(i) Determine (4 ∗ −1) ∗ 6
(ii) Is ∗ an associative binary operation?
(iii) Does Z have an identity element w.r.t ∗?
(4) Let ∗ and • be a binary operations on Z defined by x ∗ y =
∀, x, y ∈ Z .
(i) Is ∗ distributive over •?
(ii)Is • distributive over ∗?
xy
5
and x • y = xy + 5
and x • y = xy + 5
(5) Define 2 binary operations ∗ and • on Q by x ∗ y = xy
5
(i) Determine if Q is closed, associative and commutative w.r.t both operations.
(ii) Does Q have an identity and inverses w.r.t ∗ and •?
(iii) Is ∗ distributive over •?
Soln: We check if x ∗ (y • z) = (x ∗ y) • (x ∗ z)
The L.H.S is x ∗ (y • z) = x ∗ (y + 5z)
= xy+5xz
.
5
The R.H.S, (x ∗ y) • (x ∗ z) = xy
• xz
5
5
xz
= xy
+
5
= xy+5xz
, the L.H.S.
5
5
5
Thus ∗ is distributive over •.
(iv) Is • distributive over ∗?
(v) Verify if
(a) 5 • (−1 ∗ 3) = (5 • −1) ∗ (5 • 3)
(b) 5 ∗ (−1 • 3) = (5 ∗ −1) • (5 ∗ 3)
(6) Let S be the
Ç set ofå2×2 matrices over R
a b
i.e. S = {
: a, b, c, d ∈ R}, show that closure, associative, commutative,
c d
identity and inverse laws holds in R w.r.t +.
(7) Let set S = {A, B, C, D, E} with a binary operation X ∗ Y as shown on the table
below
LECTURE NOTES FOR INTRODUCTION TO ABSTRACT ALGEBRA, SMA205
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∗ A B C D E F
A B E
C A
B E A D F B C
C D F E A C
D F C B E D
E
B
F
F C
A
F
(i) Complete the table
(ii) Does associativity hold?
(iii) Is S commutative?
(iv) Determine the identity.
(v) Find the inverse for each element.
(8) For each of the following binary operations defined on Z, test for associative, commutative and inverse laws.
(i) x ∗ y = x + xy
(ii) x ∗ y = 5(x + y)
(iii) x ∗ y = x + xy + y
2.3. Relations.
Definition 2.9. A (binary) relation on a nonempty set S is a nonempty set R of ordered
pair (x, y) of elements x and y of S where x is paired with y if they satisfy the condition
of R and we usually write xRy or (x, y) ∈ R.
Definition 2.10. A relation R on a nonempty set S is an Equivalence Relation if
(i) aRa the Reflexive property
(ii) If aRb then bRa the symmetric property and
(iii) If aRb and bRcthen aRc transitivity.
Definition 2.11. We say that an integer b divides another integer a if there exists r ∈ Z
such that a = br symbolically we write b|a.
Definition 2.12. For any positive integer n and a, b ∈ Z if n divides a − b we say that a
is congruent to b modulo n and we denote it by a ≡ b mod n.
Examples
(1) Show that the relation”congruence modulo 5” is an equivalence relation.
Definition 2.13. Let R be an equivalence relation on a nonempty set S, for each a ∈ S
we define an equivalence class of a as the set of all elements of S that are related to a i.e.
[a] = {x ∈ S : xRa}
Definition 2.14. Let G be a set together a binary operation ∗ satisfying the following
properties (i) For all a, b ∈ G a ∗ b ∈ G [CLOSURE]
(ii) For all a, b and c ∈ G, a ∗ (b ∗ c) = (a ∗ b) ∗ c [ASSOCIATIVITY]
(iii) There exists a unique element e ∈ G such that for each a ∈ G, a ∗ e = e ∗ a = a
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DAMIAN M MAINGI
[IDENTITY]
(iv) For each a ∈ G there exists a0 ∈ G such that a ∗ a0 = a0 ∗ a = e [INVERSES]
is called a group.
References
[1] I N Herstein::Topics in Algebra
Springer undergraduate Mathematics series
[2] Norman Bloch:Abstract Algebra with Applications
[3] C AcholaLectures notes for undergraduate.
University of Nairobi
School of Mathematics, Kenya Science Campus, off Ngong Road, Office no 247, University of Nairobi, P.O Box 30197, 00100 Nairobi, Kenya
E-mail address: [email protected]