INTRODUCTION TO THE GROUP€¦ · INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: [email protected] Mathematics Education Study Program

Lecture Notes on Structure of Algebra

INTRODUCTION TO THE GROUP

THEORY

By :

Drs. Antonius Cahya Prihandoko, M.App.Sc

e-mail: [email protected]

Mathematics Education Study Program

Faculty of Teacher Training and Education

The University of Jember

2010

1

Acknowledgments

Thanks to the Lord, because of His Mercy, this book has been finished. This

book is written as one of the resources of the subject of Algebra Structure.

The discussion focus in this book are group and homomorphism. The descrip-

tion is started with the concepts of set and function, that are basic of all concepts

in this book. After getting a good knowledge in the concepts of set and function,

the students can continue to the chapter 2 that provides construction, properties

and order of a group and its subgroup. The next discussion on group including

cyclic group, the group of permutations and the Lagrange theorem, that will be

available in the chapter 3, 4, dan 5, respectively. The second part of the dis-

cussion is about homomorphism. This second part is started with the chapter 6

that provides homomorphism, isomorphism and Cayley theorem. Finally, the last

chapter provides the concept of factor group.

Jember, August 2010 Antonius C. Prihandoko

i

Table of Contents

Acknowledgments i

Table of Contents iii

1 SET AND FUNCTION 1

1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Partition and Equivalence Relation . . . . . . . . . . . . . . . . . 2

1.3 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Binary Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Exercise on Set and Function . . . . . . . . . . . . . . . . . . . . 6

2 GROUP 11

2.1 Definition of Group . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Properties of Group . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Order of Group and Element . . . . . . . . . . . . . . . . . . . . . 13

2.4 Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 The Exercise for Concepts of Group . . . . . . . . . . . . . . . . . 17

3 CYCLIC GROUP 20

3.1 Concept and Basic Properties . . . . . . . . . . . . . . . . . . . . 20

3.2 Subgroup of Finite Cyclic Group . . . . . . . . . . . . . . . . . . 22

3.3 Exercises on Cyclic Group . . . . . . . . . . . . . . . . . . . . . . 22

4 GROUP OF PERMUTATIONS 25

4.1 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

ii

4.2 Orbit and Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Exercises on Group of Permutation . . . . . . . . . . . . . . . . . 29

5 COSET AND THE LAGRANGE THEOREM 31

5.1 Coset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 The Lagrange Theorem . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Exercises on Coset and the Lagrange Theorem . . . . . . . . . . . 33

6 GROUP HOMOMORPHISM 36

6.1 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.2 Isomorphism dan Cayley Theorem . . . . . . . . . . . . . . . . . . 38

6.3 Exercises on group Homomorphism . . . . . . . . . . . . . . . . . 40

7 THE FACTOR GROUP 44

7.1 Factor Group Development by a Homomorphism . . . . . . . . . . 44

7.2 Factor Group Development by Normal Subgroup . . . . . . . . . . 44

7.3 The Fundamental Homomorphism Theorem . . . . . . . . . . . . 45

7.4 Exercises on the Factor Group . . . . . . . . . . . . . . . . . . . . 46

iii

Chapter 1

SET AND FUNCTION

This chapter provides the materials needed to reach the main sub-stances: (group and homomorphism), in the subject of Structure ofAlgebra. As a group is basically a set and a homomorphism is a func-tion, it is important to describe set theory and function first beforediscussing the main contents of this subject. The aim of this chapteris that the students have understanding on set, function, partition,equivalence relation and binary operation. Outcome of this chapteris that the students are able to

1. solve the set operation;

2. show an equivalence relation;

3. show a partition of a set;

4. construct a function;

5. show a binary operation;

1.1 Set

Not all concepts in mathematics can be well defined, sometime a concept can be

understood by identifying its properties. The concept of set for instance, if set

is identified as ”group of certain objects”, then there will be a question about

definition of group. If group is identified as ”unity of things”, then there will be a

question about definition of unity. This sequence of questions will be unstopped,

or we will repeat the words in previous definition. Therefore, in this chapter, set

will not be defined, but it will be identified by analyzing its characteristics.

Briefly, several things related to set can be described below.

1

I. Set and Function antonius cp 2

1. A set S consists of elements, and if a is an element of S, then it can be

notated as a ∈ S.

2. There is exactly one set that has no element. It is called empty set, dan be

notated as φ.

3. A set can be described by identifying its properties, or by listing its ele-

ments. For example, set of prime numbers less than or equal to 5, can be

described as {2, 3, 5}, or {x|x primes ≤ 5}.

4. A set is called well-defined, if it can be decided definitely whether an object

is element or not. Let S = {some natural numbers }, then S is not well-

defined set because it can not be decided whether 5 ∈ S or 5 6∈ S. If

S = {the first four natural numbers}, then elements of S can be definitely

described, that is 1, 2, 3, 4.

Definition 1.1.1 A set B is subset of set A and be notated ”B ⊆ A” or ”A ⊇B”, if every element of B is also element of A.

Note : For every set A, Both of A and φ are subset of A. A is improper subset,

while the others are proper subset.

Example 1.1.1 Let S = {a, b, c}, then S has 8 subsets that is φ, {a}, {b}, {c},{a, b}, {a, c}, {b, c}, {a, b, c}.

The concept of subset can be used to prove the equality of two sets, i.e. two

sets A and B are same if A ⊆ B and B ⊆ A.

1.2 Partition and Equivalence Relation

Definition 1.2.1 A partition of set A is a family set consisting of disjoint non

empty subsets which is union of them constructs the set A

Example 1.2.1 {{a, b}, {c}} is a partition of set S = {a, b, c}.


Based on that definition, proving that a family set {A1, A2, A3, ..., An} is a

partition of set A, can be shown below :

1. ∀i, j ∈ {1, 2, 3, ..., n}, if i 6= j then Ai ∩ Aj = φ;

2. ∪ni=1Ai = A

Example 1.2.2 1. The integers set, Z, can be partitioned to set of odd inte-

gers and set of even integers.

2. Z can be partitioned into classes of residues.

Another concept closely related to the partition is equivalence relation. If a

set is partitioned then there is an equivalence relation that can be found in that

set. Vice versa, if an equivalence relation is applied to a set, then the set of all

equivalence classes forms a partition on the set.

Definition 1.2.2 A relation ”∼” on a set A is an equivalence relation if and

only if it is:

1. reflexive; i.e ∀x ∈ A, x ∼ x;

2. symmetric; i.e if x ∼ y then y ∼ x;

3. transitive; i.e if x ∼ y and y ∼ z then x ∼ z.

Example 1.2.3 1. Relation ”same as” in the real set, <, is an equivalence

relation.

2. Let on the rational numbers set, Q, we define a relation: a/b ∼ c/d if and

only if ad = bc, then ”∼” is an equivalence relation.

3. If on the integers set, Z, we define a relation:

x ∼ y if and only if xy ≥ 0,

then ∼ is not an equivalence relation. Show it!


4. If Z be partitioned, then the relation ”is in a same partition class with” is

an equivalence relation on Z.

Definition 1.2.3 Let ”∼” is an equivalence relation on a set A, and x is an

element of A. Set of all elements equivalent to x is called equivalence class of

x, and be denoted by [x]. It is formally written as

[x] = {a ∈ A|a ∼ x}

.

Theorem 1.2.1 If x ∼ y then [x] = [y].

Prove the theorem above using the principal of the equality of two sets, i.e it has

to be shown that [x] ⊆ [y] and [y] ⊆ [x].

Theorem 1.2.2 If ”∼” is an equivalence relation on a set A, then the set of all

equivalence classes, {[x]|x ∈ A}, forms a partition on A.

For every n ∈ Z+ there is an important equivalence relation on Z that is

called as congruence modulo n.

Definition 1.2.4 Let a and b be two integers on Z and n be any positive integer.

a is congruent to b modulo n, and be denoted a ≡ b (mod n), if a − b is evenly

divisible by n, so that a − b = nk, for some k ∈ Z. Equivalence classes for

congruence modulo n are residue classes modulo n.

Example 1.2.4 We see that 7 ≡ 12 (mod 5) since 7 − 12 = 5(−1). Residue

class containing 7 and 12 is {5n + 2|n ∈ Z} = {...,−13,−8,−3, 2, 7, 12, 17, ...}

1.3 Function

Definition 1.3.1 A function φ from a set A to a set B is a relation that assigns

all elements of A into exactly one element of B.


As notation, φ : A → B is a function if (∀a ∈ A)(∃!b ∈ B), φ(a) = b. Therefore,

to show that a relation is a function, it needs to be proved that

(∀a1, a2 ∈ A), a1 = a2 =⇒ φ(a1) = φ(a2)

Definition 1.3.2 A function from a set A to a set B is injective if for each

element of B there is at most one element of A that be connected to it; and is

surjective if for each element of B there are at least one element of A that be

connected to it.

Technically, proving on that two kinds of function can be described below.

1. To show that φ is injective, it has to be proved that φ(a1) = φ(a2) implies

a1 = a2.

2. To show that φ is surjective, it has to be proved that for every b ∈ B, there

exist a ∈ A such that φ(a) = b.

1.4 Binary Operation

Definition 1.4.1 Binary operation on a set, S, is a function that assigns each

ordered pair of elements of S, (a, b) to an element of S.

This definition show that the set S has to be closure under a binary operation.

It means that if a, b ∈ S and ∗ is binary operation on S such that a ∗ b = c then

c ∈ S. Besides that the term ordered pairs has an important role, since element

that be connected to (a, b) is not necessary same as the element that be connected

to (b, a).

Example 1.4.1 1. Addition and multiplication on the set (<), on the set (Z),

on the set (C), or on the set (Q) are binary operation.

2. Let M(<) be the set of all matrices with real entries, then usual matrix

addition is not a binary operation. Why?


3. The usual addition is not a binary operation on the set <∗ = <−{0}. Why?

For identifying a binary operation ∗ on a set S, there are two things should be

considered, that is for every ordered pairs (a, b) in S,

1. ∃!c such that a ∗ b = c;

2. c ∈ S.

Definition 1.4.2 A binary operation ∗ on a set S is commutative if and only

if a ∗ b = b ∗ a, ∀a, b ∈ S; and associative if and only if (a ∗ b) ∗ c = a ∗ (b ∗ c),

∀a, b, c ∈ S.

Example 1.4.2 1. Addition and multiplication on the set (<), on the set (Z),

on the set (C), or on the set (Q) are commutative and associative.

2. Let M2,2(<) is the set of all 2 × 2 matrices with real entries, then matrix

multiplication on M2,2(<) is associative but not commutative. Please inspect

this!

3. Substraction on < is not associative and also not commutative. Why?

1.5 Exercise on Set and Function

1. If A = {1, 2, 3, 4}, how many subsets of A? Mentione it!

2. Prove that:

(a) If M ⊂ φ, then M = φ.

(b) If K ⊂ L, L ⊂ M dan M ⊂ K, then K = M .

(c) A ⊂ (A ∪B)

(d) If A ∪B = φ then A = φ dan B = φ.

(e) (A ∩B) ⊂ A

(f) A ⊂ B if and only if (A ∪B) = B

(g) (A−B) ⊂ A


(h) (A−B) ∩B = φ

(i) M ⊂ N if and only if M −N = φ

(j) M = N if and only if M −N = φ dan N −M = φ

3. Describe all elements of the following sets.

(a) {x ∈ <|x2 = 3}

(b) {a ∈ Z|a2 = 3}

(c) {x ∈ Z|xy = 60 for some y ∈ Z}

(d) {x ∈ Z|x2 − x < 115}

4. Determine whether the following relation is an equivalence relation? If yes,

describe the partition constructed by the equivalence relation!

(a) x ∼ y on the set Z if xy > 0

(b) x ∼ y on the set < if x ≥ y

(c) x ∼ y on the set < if |x| = |y|

(d) x ∼ y on the set < if |x− y| ≤ 3

(e) x ∼ y on the set Z+ if x and y have the same digits.

(f) x ∼ y on the set Z+ if x and y have the same last digit.

(g) x ∼ y on the set Z+ if n−m is divisible by 2.

5. Let n be any integer in Z+. Show that the congruence modulo n is an

equivalence relation on Z.

6. Describe all residue classes on Z modulo n, for n = 1,2,3,4 or 8

7. Compute all possible partition on a set S containing 1,2,3,4 or 5 elements.

8. If function f : A → B has inverse function f−1 : B → A, mention all

properties of f .

9. If A = [−1, 1] and function f1(x) = x2, f2(x) = x3, f3(x) = sin x, f4(x) =

x5, f5(x) = φx, observe which function that has inverse function!


10. Prove that f : A → B and g : B → C have inverse function f−1 : B → A

and g−1 : C → B, then function composition g ◦ f : A → C is also has

inverse function f−1 ◦ g−1 : C → A

11. Let f : A → B and g : B → A and g ◦ f = IA, where IA is the identity

function on A. Determine whether the following statement is true or false.

(a) f is injective.

(b) g is injective.

(c) g = f−1

(d) g is surjective.

(e) f is surjective.

12. Determine whether the following binary operator ∗ is commutative or as-

sociative.

(a) ∗ defined on Z by a ∗ b = a− b

(b) ∗ defined on Q by a ∗ b = ab + 1

(c) ∗ defined on Q by a ∗ b = ab2

(d) ∗ defined on Z+ by a ∗ b = 2ab

(e) ∗ defined on Z+ by a ∗ b = ab

13. Let a set S has exactly one element. How many different binary operation

can be defined on S? Answer the question if S has exactly 2 elements;

exactly 3 elements; exactly n elements.

14. Determine whether the following operation ∗ is binary or not. If not, de-

scribe which axiom that is not covered.

(a) On the set Z+, define ∗ by a ∗ b = a− b.

(b) On the set Z+, define ∗ by a ∗ b = ab.

(c) On the set R, define ∗ by a ∗ b = a− b.


(d) On the set Z+, define ∗ by a ∗ b = c, where c is the smallest integer

greater than botha and b.

(e) On the set Z+, define ∗ by a ∗ b = c, where c is at least 5 more than

a + b.

(f) On the set Z+, define ∗ by a ∗ b = c, where c is the largest integer less

than ab.

15. Determine the following true or false and give the reason.

(a) If ∗ is a binary operation on a set S, then a ∗ a = a,∀a ∈ S.

(b) If ∗ is a commutative binary operation on a set S, then ∀a, b, c ∈ S,

a ∗ (b ∗ c) = (b ∗ c) ∗ a.

(c) If ∗ ia an associative binary operation on a set S, then ∀a, b, c ∈ S,

a ∗ (b ∗ c) = (b ∗ c) ∗ a.

(d) A binary operation ∗ in a set S is commutative if there exists a, b ∈ S,

such that a ∗ b = b ∗ a.

(e) Every binary operation defined on a set with exactly 1 element is

commutative and associative.

(f) A binary operation on a set S assigns at least 1 element of S to each

ordered pair of elements of S.

(g) A binary operation on a set S assigns at most 1 element of S to each


(h) A binary operation on a set S assigns exactly 1 element of S to each


16. Show that if ∗ be commutative and associative binary operation on a set

S, then (a ∗ b) ∗ (c ∗ d) = [(d ∗ c) ∗ a] ∗ b, ∀a, b, c, d ∈ S.

17. Determine the following true or false and give the reason.

(a) Every binary operation on a set with one element, is commutative and

associative.


(b) Every commutative binary operation on a set with exactly 2 elements,

is associative.

(c) If F is the set of all real function, then function composition on F is

commutative.

(d) If F is the set of all real function, then function composition on F is

associative.

(e) If F is the set of all real function,, then function addition on F is

associative.

(f) If ∗ dan ∗′ be any binary operation on a set S, then a ∗ (b ∗′ c) =

(a ∗ b) ∗′ (a ∗ c), ∀a, b, c ∈ S

Chapter 2

GROUP

This chapter provides opening material of the first part of the sub-stance of the subject of Algebra Structure (also be known as AbstractAlgebra). It contains the definition, properties, order, and concepts ofgroup and its elements.

2.1 Definition of Group

An algebra structure is a system that consists of two components, that is a set

and a binary operation defined on the set. A system that consists of a non empty

set G and a binary operation ∗ defined on the set is called grupoid. If the binary

operation on the grupoid is associative, then the system is called by semi group.

If the semi group consists of identity element, e, such that for every a ∈ G satisfy

a∗e = e∗a = a, then the system is called monoid. And if each element of monoid

has own inverse, that is for each a ∈ G, ∃a−1 ∈ G such that a∗a−1 = a−1 ∗a = e,

then the system is now calledgroup. Now we go the formal definition of group .

Definition 2.1.1 A group < G, ∗ > is a set G, together with a binary operation

∗ on G, such that the following axioms are satisfied:

1. G is closed under the operation ∗. That is, ∀a, b ∈ G, a ∗ b ∈ G.

2. The binary operation ∗ is associative, that is, (∀a, b, c ∈ G), (a ∗ b) ∗ c =

a ∗ (b ∗ c).

11

II. Group antonius cp 12

3. There is an identity element, e, in G. That is, (∃e ∈ G), (∀a ∈ G),

a ∗ e = e ∗ a = a.

4. Each element of G has own inverse, that is (∀a ∈ G), (∃a−1 ∈ G), a∗a−1 =

a−1 ∗ a = e, where e is the identity element in G.

Example 2.1.1 1. The set < under usual addition operation, form a group.

2. Z5 = {0, 1, 2, 3, 4} under addition operation modulo 5, form a group.

3. {a + b√

3|a, b ∈ Z} under addition defined as follow: (a1 + b1

√3) + (a2 +

b2

√3) = (a1 + a2) + (b1 + b2)

√3, form a group.

4. The set of all 2× 2 real matrices can not form a group under matrix mul-

tiplication. Why?

5. The set Z can not form a group under multiplication. Why?

6. The set Q under multiplication, form a group.

7. Define the operation ∗, so that G = {a, b, c, d} forms a group.

Definition 2.1.2 A group < G, ∗ > is commutative if (∀a, b ∈ G), a∗b = b∗a.

Example 2.1.2 The set of integers forms a commutative group under usual ad-

dition operation.

2.2 Properties of Group

On understanding of the concepts of group, this section provides some basic

properties of group. The proof of some theorems will be left as exercise.

Theorem 2.2.1 The identity element in a group is unique.

Proof: If identity element is not unique then there are e′ and e and both of them

are identity elements. If e′ as the identity element then e ∗ e′ = e. If e as the

identity element then e ∗ e′ = e′. Since ∗ is a binary operation then e′ = e.


Theorem 2.2.2 The inverse of each element of a group is unique.

Proof: If the inverse of a is not unique, then there are two different elements b

and c, where both of them are inverse of a. So that (b∗a)∗c = c dan b∗(a∗c) = b.

Thus, b = c.

Theorem 2.2.3 If G is a group under binary operation ∗, then G satisfies the

left cancelation law and the right cancelation law. That is, a ∗ b = a ∗ c implies

b = c, and a ∗ b = c ∗ b implies a = c, ∀a, b, c ∈ G.

Proof for the left cancelation law:

a ∗ b = a ∗ c =⇒ a−1 ∗ a ∗ b = a−1 ∗ a ∗ c

=⇒ e ∗ b = e ∗ c

=⇒ b = c

Using the analog way, prove the right cancelation law!

Theorem 2.2.4 If G is a group and a1, a2, · · · , an is any n elements of G, then

(a1 ∗ a2 ∗ · · · ∗ an)−1 = a−1n ∗ a−1

n−1 ∗ · · · ∗ a−11

.

Theorem 2.2.5 If G is a group then for all element a in G (a−1)−1 = a.

Theorem 2.2.6 In a group G, the equation ax = b, where a, b ∈ G dan x is a

variable, has unique solution, that is x = a−1b.

Theorem 2.2.7 If an empty set G under binary operation ∗ satisfy the axioms:

closed, associatif, and the equation a ∗x = b and y ∗ a = b have solution for every

a, b ∈ G, then (G, ∗) is a group.

2.3 Order of Group and Element

This section provides the definition of order group and elements and its properties.


Definition 2.3.1 The result of operation of m factors, a ∗ a ∗ a ∗ a ∗ · · · ∗ a is

represented by am; The result of operation of m factors, a−1∗a−1∗a−1∗a−1∗· · ·∗a−1

is represented by a−m; and a0 = e, where e is the identity element in G.

Theorem 2.3.1 If m is a positif integer then a−m = (a−1)m = (am)−1

Example 2.3.1 1. In the group (Z, +), 47 = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28;

4−1 = −4 so that 4−5 = (−4) + (−4) + (−4) + (−4) + (−4) = −20; 40 = 0,

since 0 is the identity element under addition of integers.

2. In the group (<,×), 23 = 2×2×2 = 8; 2−1 = 12

so that 2−3 = 12× 1

2× 1

2= 1

8;

20 = 1, since 1 is the identity element under multiplication of real numbers.

Theorem 2.3.2 If m and n are integers then am ∗ an = am+n and (am)n = amn

Definition 2.3.2 The order of a finite group G is the number of elements of

G. If the number of elements of G is infinite, then the order of G is infinite. The

order of G is denoted as |G|.

Definition 2.3.3 Let a is an element of a group G. The order of a is n if and

only if n is the least positive integer such that an = e, where e is identity element

in group G. If there is no such positive integer, then the order of a is infinite.

The order of a is denoted O(a).

Example 2.3.2 1. In the group (Z5, +), O(2) = 5, since 5 is the least positive

integer such that 25 = 2 + 2 + 2 + 2 + 2 ≡ 0(mod5).

2. In the group (Z5,×), O(2) = 4, since 4 is the least positive integer such

that 24 = 2× 2× 2× 2× 2 ≡ 1(mod5).

Theorem 2.3.3 Let a be an element of a group G. If the order of a is n then

there exist n variation of power of a in G, they are a1, a2, a3, · · · , an−1, an

Note that the power here depends on the defined binary operation.

Example 2.3.3 1. In (Z5, +), O(2) = 5, so that there exist 5 variation of

power of 2, that is 21 = 2; 22 = 4; 23 = 1; 24 = 3; dan 25 = 0.


2. In (Z5,×), O(2) = 4, so that there exist 4 variation of power of 2, that is

21 = 2; 22 = 4; 23 = 3; dan 24 = 1.

Theorem 2.3.4 If the order of a is infinite then all power of a are distinct, that

is if r 6= s than ar 6= as.

Example 2.3.4 In the (Z, +), O(2) is infinite, so that every power of 2 is dif-

ferent.

Prove the following two theorem and give an example for each.

Theorem 2.3.5 Let O(a) = n. (ak = e) ⇔ n|k (n is a factor of k).

Theorem 2.3.6 Let O(a) = n then O(a−1) = n

2.4 Subgroup

Definition 2.4.1 Let < G, ∗ > be a group and H be a non empty subset of G.

H is a subgroup of G if and only if < H, ∗ > is also a group.

Base on the definition, a subgroup is a group inside the other group. Fur-

thermore, since H is a subset of G then the axiom associative also works on

H.

Theorem 2.4.1 Let < G, ∗ > be a group and H be a non empty subset of G. H

is a subgroup of G if it satisfy these three axioms.

1. Closed.

2. Identity element

3. Inverse.

It also can be analyzed that if the system satisfy axiom closed and inverse, than

the identity element will be automatically satisfied. Therefore the theorem above

can be simplify to the following theorem.


Theorem 2.4.2 Let < G, ∗ > be a group and H be non empty subset of G. H

is a subgroup of G if it satisfy the two axioms below.

1. Closed. That is (∀c, d ∈ H), c ∗ d ∈ H.

2. Inverse. That is (∀c ∈ H), c−1 ∈ H.

Finally, those two axioms in above theorem can be combined to achieve the

following theorem.

Theorem 2.4.3 Let < G, ∗ >be a group and H be a non empty subset of G. H

is a subgroup of G if (∀c, d ∈ H), c ∗ d−1 ∈ H.

Example 2.4.1 Both of {0, 3} and {0, 2, 4} are subgroup of Z6, +). Show it!

Definition 2.4.2 Let < G, ∗ > be a group. Both of H and K are subset of G.

Then

H ∗K = {a ∈ G|a = h ∗ k, h ∈ H ∧ k ∈ K}

and

H−1 = {a ∈ G|a = h−1, h ∈ H}

Those definition can be used on proving the next theorems.

Theorem 2.4.4 If < H, ∗ > is a subgroup of a group < G, ∗ >, then H ∗H = H

and H−1 = H.

Theorem 2.4.5 If both of H and K are subgroup of a group < G, ∗ >, then

H ∗K is also a subgroup if and only if H ∗K = K ∗H.

Theorem 2.4.6 If both of H and K are subgroup of a group (G, ∗), then H ∩K

is also a subgroup on < G, ∗ >.

Theorem 2.4.7 Let G be a group and a ∈ G. If H is the set of all power of a

in G, then H is a subgroup of G.

Example 2.4.2 In the (Z5,×), there are 4 variation of power of 2, that is 21 = 2;

22 = 4; 23 = 3; and 24 = 1, so that {1, 2, 3, 4} is a subgroup of (Z5,×).


2.5 The Exercise for Concepts of Group

1. Determine whether the following algebra structure is a grupoid, semigrup,

monoid or group!

(a) The set of natural numbers under usual addition.

(b) The set of natural numbers under usual multiplication.

(c) The set of integers under usual addition.

(d) G = {ma|a ∈ Z} under integers addition, where m is any integer.

(e) G = {ma|a ∈ Z} under multiplication, where m is any integer.

(f) G = {a +√

2b|a, b ∈ Q} under addition.

(g) The set of non zero rational numbers under multiplication.

(h) The set of non zero complex numbers under multiplication.

(i) {0, 1, 2, 3, ...} under addition.

(j) The set M2(<) under addition matrix.

(k) The set M2(<) under multiplication matrix.

(l) The set of all integers divisible by 5, under addition.

(m) The set of all vectors in <2 of the form (x, 3x) under addition vector.

(n) The set of all vectors in <2 of the form (0, y) or (x, 0) under addition

vector.

(o) G = {f1, f2, f3, f4} under transformation composition, where f1(z) =

z, f2(z) = −z, f3(z) = 1z, f4(z) = −1

z, for every complex numbers, z.

2. Let S be the set of all real numbers except −1. On S define ∗ by a ∗ b =

a + b + ab.

(a) Show that ∗ is a binary operation.

(b) Show whether S is a group or not.

(c) Compute the solution of 2 ∗ x ∗ 3 = 7 in S.


3. If G is a group with binary operation ∗, show that ∀a, b ∈ G, (a ∗ b)−1 =

b−1 ∗ a−1!

4. Determine true or false the following statements and give the reason.

(a) A group may has more than one identity element.

(b) On a group, every linear equation has solution.

(c) Every finite group containing at most 3 elements is abelian.

(d) The empty set can be considered as a group.

5. If G is a finite group with identity element e and of even order, show that

there exists a 6= e in G such that a ∗ a = e!

6. Let ∗ be a binary operation on a set S, an element x ∈ S is called idempotent

for ∗ if x ∗ x = x. Show that a group has exactly one idempotent element.

7. If G is a group with identity element eand ∀x ∈ G, x ∗ x = e, show that G

is an abelian group.

8. Show that if (a ∗ b)2 = a2 ∗ b2, for a and b in G, then a ∗ b = b ∗ a!

9. Let G be a group and a, b ∈ G. Show that (a ∗ b)−1 = a−1 ∗ b−1 if and only

if a*b = b*a!

10. Determine true or false the following statements and give the reasons.

(a) Associative law is always satisfied on every group.

(b) There may exist a group which is not satisfy the cancelation law.

(c) Every group is a subgroup of itself.

(d) Every group has exactly two improper subgroup.

(e) Every set of numbers that is a group under addition, is also a group

under multiplication.

(f) Every subset of a group is a subgroup under the same binary operation.

(g) If b2 = e then b = e, where e is the identity element.


(h) If c2 = c then c = e.

(i) On every group, an ∗ bn = (a ∗ b)n.

11. If both of H and K are subgroup of an abelian group G, show that HK =

{hk|h ∈ H, k ∈ K} is also subgroup of G.

12. Show that a non empty subset H of a group G is a subgroup of G if and

only if ab−1 ∈ H, ∀a, b ∈ H!

13. Show that if G is an abelian group with identity element e, then all element

x ∈ G satisfying the equation x2 = e forms a subgroup H of G!

14. Let G be a group and a is an element in G. Show that Ha = {x ∈ G|xa =

ax} is a subgroup of G!

15. Let H be a subgroup of a group G. For a, b ∈ G, let a ∼ b if and only

if ab−1 ∈ H. Show that ∼ is an equivalence relation on G. Describe the

partition formed by such equivalence relation.

16. Show that if H is a subgroup of G and K is a subgroup of G, then H ∩K

is also a subgroup of G!

Chapter 3

CYCLIC GROUP

As discussed before that the set of all power of an element of a groupforms a subgroup. It derives the concept of a group containing allpower of an element in it. Such group is then called cyclic group.

3.1 Concept and Basic Properties

In the end of previous chapter, it has been described that the set of all power

of an element in group forms subgroup. Let G be a group and a ∈ G, then

H = {h ∈ G|h = ak, k ∈ Z} is a subgroup of G. Therefore a is also in H since

a = a1 and all elements of H can be presented as a power of a. It can be said

that a generates the set H, which is a group. This concept is a basic of forming

cyclic group.

Definition 3.1.1 A group G is cyclic if there exists elements a ∈ G such that

every element x ∈ G, can be represented by x = am, where m is integer. The

element a is called by generator and G is a cyclic group developed by a and

denoted :

G =< a >

.

Theorem 3.1.1 Every cyclic group is abelian.

Theorem 3.1.2 If G =< a > and b ∈ G then O(b)|O(a).

20

III. Cyclic Group antonius cp 21

Proving on the next theorems of cyclic group need division algorithm on in-

tegers (Z). This algorithm is based on the theorem that if b is a positive integer

and a be any integer, then there exists unique integers q dan r such that

a = bq + r

where 0 ≤ r < b.

Theorem 3.1.3 Every subgroup of cyclic group is cyclic.

Corollary 3.1.1 Subgroups of Z under addition are precisely the groups nZ un-

der addition for n ∈ Z.

Definition 3.1.2 Let r and s be two positive integers. The positive generator

/(d/) of the cyclic group

G = {nr + ms|n,m ∈ Z}

under addition is the greatest common divisor (gcd) of r and s.

To understanding that definition, firstly we have to show that G is a subgroup

of Z. This is easy to do. Since (Z, +) =< 1 > then G is also cyclic and has a

positive generator d. Note from the definition that d is a divisor of r and s since

both r = 1r + 0s and s = 0r + 1s are in G. Since d ∈ G, then it can be written

as

d = nr + ms

for any integers n and m. It can be shown that every integer that divides r and

s will also divides d. Therefore d is the greatest common divisor of r and s.

Theorem 3.1.4 If G =< a > is of order non prime n, then every proper sub-

group of G is generated by am where m is proper divisor of n. In converse, if m

is a proper divisor of n then G has a proper subgroup generated by am.

Theorem 3.1.5 If G =< a > and O(a) = n then |G| = n.

Theorem 3.1.6 If G =< a > then G =< a−1 >.


3.2 Subgroup of Finite Cyclic Group

Let G =< a >, then G may be finite or infinite.

1. If G is infinite, then all power of a are different. Prove it!

2. If G is finite and of order n, then there are exactly n different power of a.

Why?

Definition 3.2.1 Let n be a fixed positive integer and let h and k be any integers.

The remainder r when h+ k is divided by n in accord with the division algorithm

is the sum of h and k modulo n.

Example 3.2.1 13 + 18 = 31 = 5(6) + 1. So that 13 + 18 ≡ 1 (mod 5).

Theorem 3.2.1 The set {0, 1, 2, 3, · · · , n− 1} is the cyclic group Zn under ad-

dition modulo n.

Theorem 3.2.2 Let G =< a > and |G| = n. If b ∈ G and b = as, then b

generate a cyclic subgroup H of G containing nd

elements, where d is gcd of n

and s.

Example 3.2.2 Consider Z12 with generator a = 1. Since8 = 8 · 1 and gcd of

12 and 8 is 4 then 8 generate a subgroup containing 124

= 3 elements, that is

< 8 >= {0, 4, 8}.

Theorem 3.2.3 If G =< a > and |G| = n then the other generators for G are

of the forms ar, where r is relatively prime to n.

Example 3.2.3 as we know that Z12 =< 1 >. Then the other generators for

Z12 are 5 = 5 · 1, 7 = 7 · 1, dan 11 = 11 · 1.

3.3 Exercises on Cyclic Group

1. Prove that every cyclic group is abelian!

2. Show that a group having no proper non trivial subgroup is a cyclic group!


3. Compute the gcd of 32 and 24, 48 and 88, 360 and 420.

4. Let +n be notation of addition modulo n. Compute: 13 +17 8, 21 +30 19,

26 +42 16, and 39 +54 17.

5. Compute the number of generator of cyclic group of order : 5, 8, 12, and

60.

6. Compute the order of a cyclic subgroup of Z30 generated by 25.

7. Find all subgroup of : Z12, Z36, and Z8, and draw their lattice diagram.

8. Determine all possibilities order of subgroups of the group : Z6, Z8, Z12,

Z60, and Z17.

9. Determine true or false the following statement and give the reason!

(a) In every cyclic group, every element is a generator.

(b) Z4 is a cyclic group.

(c) Every abelian group is cyclic.

(d) Q under addition is cyclic.

(e) Every element of cyclic group is generator.

(f) There is at least one abelian group of every finite order > 0.

(g) All generators of Z20 are prime numbers.

(h) Every cyclic group of order > 2 has at least two distinct generators.

10. Give an example of a group with the property described, or explain why no

example exists.

(a) A finite group that is not cyclic.

(b) An infinite group that is not cyclic.

(c) A cyclic group having only one generator.

(d) An infinite cyclic group having four generators.

(e) A finite cyclic group having four generators.


11. If G =< a > and o(a) = n, prove that |G| = n.

12. If G =< a >, prove that G =< a−1 >.

13. If a group of order n containing an element of order n, prove that the group

is cyclic.

14. In a cyclic group of order n, show that there exists element of order k, where

k is a factor of n.

15. Show that a group having finite number of subgroup, is finite.

16. Let p and q be two prime numbers. Compute the number of generators of

cyclic group Zpq.

17. Let p be a prime number. Compute the number of generators of cyclic

group Zpr , where r is an integer ≥ 1.

18. Show that in a finite cyclic group Gof order n, the equation xm = e has

exactly m solutions in G for each positive integer m that divides n.

19. Show that Zp has no proper subgroup if p is a prime number.

20. Prove that the order of a cyclic group same as the order of its generator.

21. How many generators of a cyclic group of order 10?

Chapter 4

GROUP OF PERMUTATIONS

In this chapter we describe a group containing permutations definedin a set. Such group has special characteristics.

4.1 Permutation

Definition 4.1.1 Permutation in a set A is a one to one function from A onto

A.

Example 4.1.1 1. Let A = {x, y, z} then

α =

a b c

c a b

is a permutation, where α(a) = c; α(b) = a; and α(c) = b.

2. Let B = {1, 2, 3, 4, 5}. Given two permutations in B,

φ =

1 2 3 4 5

1 2 5 3 4

and

β =

1 2 3 4 5

4 1 2 5 3

then permutation multiplication (= function composition) φβ is :

φβ =

1 2 3 4 5

1 2 5 3 4

1 2 3 4 5

4 1 2 5 3

=

1 2 3 4 5

3 1 2 4 5

25

IV. Group of Permutations antonius cp 26

Theorem 4.1.1 Let A be a non empty set and SA be the set of all permutation

in A. Then SA is a group under permutation multiplication.

Definition 4.1.2 Symmetric Group. Let A = {1, 2, 3, · · · , n}, then group of

all permutations on A is called symmetric group n, and be denoted as Sn.

Note : Sn has n! elements.

Example 4.1.2 Let A = {1, 2, 3} then S3 has 3! = 6 elements. All permutations

on A can be described below.

ρ0 =

1 2 3

1 2 3

, µ1 =

1 2 3

1 3 2

,

ρ1 =

1 2 3

2 3 1

, µ2 =

1 2 3

3 2 1

,

ρ2 =

1 2 3

3 1 2

, µ3 =

1 2 3

2 1 3

,

It can be proved that S3 = {ρ0, ρ1, ρ2, µ1, µ2, µ3} is a group under permutation

multiplication.

4.2 Orbit and Cycle

Every permutation σ of a set A determines a natural partition of A into cells

with the property that a, b ∈ A are in the same cell if and only if b = σn(a), for

some n ∈ Z. It can be proved that relation ∼ defined by a ∼ b ⇔ b = σn(a), is

a equivalence relation .

Definition 4.2.1 Let σ be a permutation of a set A. The equivalence classes

determined by the equivalence relation

a ∼ b ⇔ b = σn(a)

are the orbits of σ.


Example 4.2.1 The orbits of permutation

σ =

1 2 3 4 5 6 7 8

8 1 6 3 7 4 5 2

of S8 can be found by applying σ repeatedly, obtaining symbolically

1 −→ 8 −→ 2 −→ 1

3 −→ 6 −→ 4 −→ 3

5 −→ 7 −→ 5

hence the orbits of σ are

{1, 2, 8}, {3, 4, 6}, {5, 7}

Each orbit in the example above determines a new permutation in S8 by acts

on the orbit members and leaves the remaining elements fixed. For example the

orbit {1, 2, 8} with arrow direction

1 −→ 8 −→ 2 −→ 1

forms the permutation

µ =

1 2 3 4 5 6 7 8

8 1 3 4 5 6 7 2

The permutation µ has only 1 orbit containing more than 1 element. Such per-

mutation is called cycle. Let say the formal definition.

Definition 4.2.2 A permutation σ ∈ Sn is a cycle if σ has at most one orbit

containing more than one element. The length of a cycle is the number of

elements in its largest orbit.

Example 4.2.2 As mentioned in previous example, permutation µ is a cycle of

length 3 and be denoted as

µ = (1, 8, 2)


Note, unlike in the orbit notation, the order of elements in the cycle notation

determines moving flow. For example, (1, 8, 2) = (8, 2, 1) = (2, 1, 8) but (1, 8, 2) 6=(1, 2, 8).

As described before that the set of orbits of a permutation is a partition on

Sn, therefore the orbits of a permutation are disjoint sets. Furthermore, since an

orbit determines a cycle, then we can derive the following theorem.

Theorem 4.2.1 Each permutation σ in a finite set is a product of disjoint cycles.

Example 4.2.3

σ =

1 2 3 4 5 6 7 8

8 1 6 3 7 4 5 2

= (1, 8, 2)(3, 6, 4)(5, 7)

Definition 4.2.3 A cycle of length 2 is called transposition.

Each cycle can be described as a product of transpositions,

(a1, a2, a3, · · · , an−1, an) = (a1, an)(a1, an−1) · · · (a1, a3)(a1, a2).

Therefore, a permutation is also a product of transpositions.

Theorem 4.2.2 Let σ ∈ Sn and τ be a transposition on Sn. The number of

orbits of σ and the number of orbits of τσ differ by 1.

Let τ = (i, j), then the above theorem can be proved by analyzing the possi-

bilities below.

1. i and j are in the different orbits of σ;

2. i and j are in the same orbit of σ.

Definition 4.2.4 A permutation of a finite set is even or odd according to

whether it can be expressed as a product of an even number of transposition or

the product of an odd number of transposition, respectively.

Example 4.2.4 Determine whether even or odd the following permutations.


1. σ =

1 2 3 4 5 6 7 8

8 1 6 3 7 4 5 2

2. µ =

1 2 3 4 5 6 7 8

8 1 3 4 5 6 7 2

Definition 4.2.5 The subgroup of Sn consisting even permutations is called al-

ternating group, An on n letters.

Example 4.2.5 Determine the alternating group of S3!

4.3 Exercises on Group of Permutation

1. Compute the number of elements of the set below.

(a) {α ∈ S4|α(3) = 3}.

(b) {α ∈ S5|α(2) = 5}.


(a) Every permutation is a one to one function.

(b) Every function is permutation if and only if it is one to one.

(c) Every function from a set onto itself should be one to one.

(d) The symmetric group S10 has 10 elements.

(e) The symmetric groupS3 is cyclic.

(f) Sn is not cyclic for each n.

3. Show that the symmetric group Sn is not abelian for n ≥ 3.

4. Find all orbits of the following permutations:

(a) α : Z −→ Z, where α(n) = n + 1.

(b) α : Z −→ Z, where α(n) = n + 2.

(c) α : Z −→ Z, where α(n) = n− 3.


5. For permutation on {1, 2, 3, 4, 5, 6, 7, 8}, compute the product of cycles be-

low.

(a) (1, 4, 5)(7, 8)(2, 5, 7)

(b) (1, 3, 2, 7)(4, 8, 6)

(c) (1, 2)(4, 7, 8)(2, 1)(7, 2, 8, 1, 5)

6. Express the following permutations as a product of disjoint cycles, and then

express them as product of transpositions.

(a)

1 2 3 4 5 6 7 8

8 2 6 3 7 4 5 1

(b)

1 2 3 4 5 6 7 8

3 6 4 1 8 2 5 7

(c)

1 2 3 4 5 6 7 8

3 1 4 7 2 5 8 6


(a) Every permutation is a cycle.

(b) Every cycle is a permutation.

(c) Alternating group A5 has 120 elements.

(d) Alternating group A3 is a commutative group.

(e) The symmetric group Sn is not cyclic for every n ≥ 1.

8. Show that if H is a subgroup of Sn, for n ≥ 2, then all permutation in H

is even or exactly half of them are even.

9. Let G be a group and a be a fixed element in G. Show that map λa : G −→G, where λa(g) = ag, ∀g ∈ G, is a permutation in G.

Chapter 5

COSET AND THE LAGRANGE

THEOREM

A subgroup of a group has cosets. The set of the cosets determines apartition on the group. Furthermore, there is a one to one correspon-dence between the subgroup and its coset. This concept leads therelationship between the order of a group and the order of subgroup.

5.1 Coset

Theorem 5.1.1 Let H be a subgroup of a group G. The relation ∼L determined

by

a ∼L b ⇔ a−1b ∈ H

and the relation ∼R determined by

a ∼R b ⇔ ab−1 ∈ H

are an equivalence relation.

Prove the theorem above!

The relation ∼L or ∼R determine equivalence classes in G. For example, the

class containing a formed by ∼L is [a] = {x ∈ G|a ∼L x} or equivalently equal

to {x ∈ G|a−1x ∈ H}, or equivalently equal to {x ∈ G|a−1x = h, h ∈ H}, or

equivalently equal to {x ∈ G|x = ah, h ∈ H}. So that [a] = {ah|h ∈ H} = aH.

Please observe the class containing a formed by ∼R.

31

V. Coset and the Lagrange Theorem antonius cp 32

Definition 5.1.1 Let G be a group and H be a subgroup of G. For any element

a ∈ G; aH = {x ∈ G|x = ah, h ∈ H} is called left coset of H containing a; and

Ha = {y ∈ G|y = ha, h ∈ H} is called right coset of H containing a.

Example 5.1.1 Z is a group under integer addition and 3Z= {· · · ,−6,−3, 0, 3, 6, 9, · · · }is a subgroup of Z. The left cosets of 3Z in Z are

• 3Z= {· · · ,−6,−3, 0, 3, 6, 9, · · · }

• 1 + 3Z= {· · · ,−5,−2, 1, 4, 7, 10, · · · }

• 2 + 3Z= {· · · ,−4,−1, 2, 5, 8, 11, · · · }

It can be shown that {3Z, 1 + 3Z, 2 + 3Z} is a partition of Z.

Whether we use left coset or right coset, there is no significantly different,as

long as we are consistent. In the next description we use left coset, which is

analog to the right coset. Here are some theorems regarding the coset.

Theorem 5.1.2 Coset aH = bH if and only if a ∈ bH.

Theorem 5.1.3 Let G be a group and H be a subgroup of G. The family set of

all coset of H is a partition of G.

Theorem 5.1.4 If aH is a coset of H, then H and aH are one to one corre-

spondence.

5.2 The Lagrange Theorem

If G be a group and H be a subgroup of G, then the set of all cosets of H forms

a partition on G and H is one to one corresponds to each coset. For finite set,

the concept of one to one correspondence indicates the equality of the number of

elements. Therefore, in a finite group, there is a relationship between the group

order and its subgroup order.

Theorem 5.2.1 The Lagrange Theorem. If G be a finite group and H sub-

grup of G, then the order of H is a factor of the order of G.


Example 5.2.1 1. If G is of order 8, then the possible order for H are 1, 2,

4, or 8. If |H| = 1 then H = {e}, where e is the identity element of G. If

|H| = 8 then H = G.

2. It can be observe that subgroups of Z6 = {0, 1, 2, 3, 4, 5} are {0}, {0, 3},{0, 2, 4}, dan Z6 itself.

The following are some theorems as corollary of the Lagrange theorem.

Theorem 5.2.2 If G be a prime ordered group then G is a cyclic group and each

element except the identity element is a generator.

Theorem 5.2.3 The order of each element of a finite group is a factor of the

order of the group.

Definition 5.2.1 If G be a group and H be a subgrup of G then index of H is

the number of cosets of H in G, and be denoted

(G : H)

.

Therefore, a finite group satisfies

(G : H) =|G||H|

5.3 Exercises on Coset and the Lagrange Theo-

rem

1. Find all cosets of the following subgroups.

(a) 4Z of group Z.

(b) 4Z of group 2Z.

(c) < 2 > of group Z12.

(d) < 4 > of group Z12.


(e) < 18 > of group Z36.

(f) {ρ0, ρ2} of group D4.

(g) {ρ0, µ2} of group D4.

2. Compute the index of each subgroup below.

(a) < 3 > of group Z24.

(b) < µ1 > of group S3.

(c) < µ3 > of group D4.


(a) Every subgroup of every group has coset.

(b) The number of coset of subgroup in a finite group divides the order of

the group.

(c) Every group of prime order is abelian.

(d) A subgroup is a coset for itself.

(e) There only subgroup of a finite group that has coset.

4. Let H be a subgroup of a group G such that g−1hg ∈ H, ∀g ∈ G and

∀h ∈ H. Show that each left coset of H, gH, same as the right coset, Hg!

5. Let H be a subgroup of a group G. Show that if the partition on G by the

left coset of H same as the partition on G by the right coset of H, then

g−1hg ∈ H is satisfied, ∀g ∈ G and ∀h ∈ H!

6. Give the proof if the following is true, or give acounter-example if it is false.

(a) If aH = bH then Ha = Hb

(b) If Ha = Hb then b ∈ Ha

(c) If aH = bH then Ha−1 = Hb−1

(d) If aH = bH then a2H = b2H


7. If G be a group of order pq, where both of p and q are prime, show that

each proper subgroup of G is cyclic.

8. Show that a group having at least two elements but no proper non trivial

subgroup, must be a finite group of prime order.

9. Show that if H be a subgroup of index 2 in a finite group G, then each left

coset of H is also its right coset.

10. Show that if a group G with identity elemente and of finite order, n, then

an = e, ∀a ∈ G!

11. Let H and K be two subgroup of a group G. Define a relation ∼ in G

determined by a ∼ b if and only if a = hbk, for any h ∈ H dan k ∈ K.

(a) Prove that ∼ is an equivalence relation in G!

(b) Describe all elements of the equivalence class containing a!

Chapter 6

GROUP HOMOMORPHISM

This chapter is the beginning of second part of the subject of AlgebraStructure. In the first part, we discuss about group and its properties.In this second part we will discuss about a kind of function that assignsa group to another group. This chapter provides the definition ofhomomorphism and its properties.

6.1 Homomorphism

Definition 6.1.1 A map φ from a group (G, ∗) into a group (G′, #) is called

homomorphism if

φ(a ∗ b) = φ(a)#φ(b)

for all a and b in G.

The equation in the above definition shows a relationship between two binary

operations ∗ and #, and thus a relationship between two groups G and G′.

Notes : For simplify the notation we will not write the binary operation symbol.

Let G be a group and a, b ∈ G then binary operation on a and b be written as ab.

Between any two groups G and G′, there is at least a homomorphism φ : G →G′ called trivial homomorphism defined by φ(g) = e′ for each g ∈ G where e′ is

the identity element of G′.

Example 6.1.1 Let α : Z → Zn defined by α(m) = r, where r is the remainder

when m is divided by n. Then α is a homomorphism.

36

VI. Group Homomorphism antonius cp 37

Example 6.1.2 Let Sn be a symmetric group on n letters, and let φ : Sn → Z2

defined by

φ(ρ) =

0 if ρ is an even permutation,

1 if ρ is an odd permutation

Then φ is a homomorphism. Prove it! !

Example 6.1.3 Let F be an addition group of all function in R, and R be an

addition group of real numbers, and let c be a real number. Let φc : F → R

defined by φc(f) = f(c) for each f ∈ F . Then φc is a homomorphism and is

called evaluation homomorphism.

Definition 6.1.2 Let φ be a mapping of a set X into a set Y , and let A ⊆ X and

B ⊆ Y . The image of A in Y is φ(A) = {φ(a)|a ∈ A}. The set φ(X) is called

the range of φ. The inverse image of B in X is φ−1(B) = {x ∈ X|φ(x) ∈ B}

Theorem 6.1.1 Let φ be a homomorphism of a group g into a group G′,

1. If e is the identity in G, then φ(e) is the identity e′ in G′;

2. If a ∈ G, then φ(a−1) = φ(a)−1;

3. If H is a subgroup of G, then φ(H) is a subgroup of G′;

4. If S ′ is a subgroup of G′, then φ−1(S ′) is a subgroup of G;

Loosely speaking, φ preserves the identity, inverses, and subgroups.

Definition 6.1.3 Let φ : G → G′ be a homomorphism, then the kernel of φ,

denoted by Ker(φ), is defined as

Ker(φ) = φ−1({e′}) = {a ∈ G|φ(a) = e′}

where e′ is the identity of G′.


Theorem 6.1.2 If φ : G → G′ is a homomorphism, then Ker(φ) is a subgroup

of G.

Prove it!

Theorem 6.1.3 Let φ : G → G′ be a homomorphism and H = Ker(φ). Let

a ∈ G. Then

φ−1{φ(a)} = {x ∈ G|φ(x) = φ(a)}

is the left coset aH of H and is also the right coset Ha of H. Consequently, the

two partitions of G into left cosets and into right cosets of H are the same.

Corollary 6.1.1 A group homomorphism φ : G → G′ is a one-to-one map if

and only if Ker(φ) = {e}.

Prove it!

Definition 6.1.4 One-to-one homomorphism is called monomorphism , onto

homomorphism is called epimorphism .

Definition 6.1.5 A subgroup H of a group G) is normal if gH = Hg for all

g ∈ G.

6.2 Isomorphism dan Cayley Theorem

Definition 6.2.1 An isomorphism φ : G → G′ is a one-to-one homomorphism

of G onto G′. If so then we say that G and G′ are isomorph and be denoted as

G ' G′.

Theorem 6.2.1 Let ζ be a set of groups. The relation ' in ζ is a equivalence

relation.

Prove it !

The steps to show two groups that are isomorph is as follows.

1. Define a function φ.


2. Show that φ is one-to-one function.

3. Show that φ is onto.

4. Show that φ(xy) = φ(x)φ(y), ∀x, y ∈ G.

Theorem 6.2.2 All infinite cyclic group, G, are isomorph to the group of inte-

gers, Z, under addition.

Prove the above theorem by define a function φ : G → Z where φ(an) = n,

∀an ∈ G.

Now, how to show that two groups are not isomorph? To do that we need to

know some structural properties, that is the properties that must be shared by

any isomorphic groups. Some examples on the structural properties are

• cyclic;

• abelian;

• finite and infinite;

• group order;

• the number of elements with certain order;

• the solution of an equation on a group.

The non structural properties, in other hand, may not be shared by any

isomorphic groups. Some example of them are as follows.

• group containing 5;

• all elements of group are number;

• the binary operation on a group is a function composition;

• the elements of a group are permutations;

• the group is a subgroup of < R, >


Example 6.2.1 1. It cannot be said that Z and 3Z are not isomorph since

17 ∈ Z but 17 6∈ 3Z. This properties is non structural properties. In

the fact Z and 3Z are isomorph since the function φ : Z → 3Z, where

φ(n) = 3n, is a isomorphism.

2. It can not be said that Z and Q are not isomorph since 12∈ Q but 1

26∈ Z.

But it can be said that Z and Q are not isomorph since Z is cyclic, but Qis not cyclic.

Theorem 6.2.3 Cayley Theorem. Every group isomorphs to a group of per-

mutation.

Prove the Cayley theorem above by following steps.

1. Beginning with a group G, collect all permutation in G into a set G′

2. Prove that G′ is a group under permutation multiplication.

3. Define function φ : G → G′ and show that φ is an isomorphism of G to the

G′.

6.3 Exercises on group Homomorphism

1. Determine whether the following mapping is a homomorphism or not and

give the reasons.

(a) φ : Z → < defined by φ(x) = x.

(b) φ : < → Z defined by φ(x) = bxc.

(c) φ : Z6 → Z2 defined by φ(x) = the remainder when x is divided by 2.

(d) φ : Z9 → Z2 defined by φ(x) = the remainder when x is divided by 2.

(e) φ : Z → Z defined by φ(x) = −x

(f) φ : F → F defined by φ(f) = 3f , where F is the group of all real

functions.

(g) φ : < → Z under addition, and be defined by φ(x) = bxc, ∀x ∈ <.


(h) φ : [<, +] → [<, ·] defined by φ(x) = 2x

(i) φi : Gi → G1×G2×...×Gi×...×Gr defined by φi(gi) = (e1, e2, ..., gi, ..., er),

dimana gi ∈ Gi dan ej elemen identitas dalam Gj.

(j) φ : G → G defined by φ(x) = x−1, ∀x ∈ G.

(k) φ : F → F defined by φ(f) = f”, ∀f ∈ F , where F is the addition

group of all functions << that are differentiable on all order.

(l) φ : F → R defined by φ(f) =∫ 4

0f(x)dx, where F is the addition group

of all continue function <<; < is the addition group of real numbers.

(m) φ : F → F defined by φ(f) = 3f , where F is the addition group of all

function <<.

(n) φ : Mn → < defined by φ(A) = det(A), where Mn the addition group

of all matrices 2x2, and < is the addition group of all real numbers.

2. Let F = << be an addition group, D = {f ∈ <<|f differentiable }. Is

φ : f ∈ D → f ′ ∈ F homomorphism or not? If yes, find Ker(φ).

3. Let G be a group. If φ : Z×Z → G is a homomorphism and let φ(1, 0) = h

and φ(0, 1) = k, compute φ(m, n).

4. Let G be a group and g ∈ G. Let φg : x ∈ G → gx ∈ G. Determine g, in

orderφg be homomorphism.

5. Let G be a group and g ∈ G. Let φg : x ∈ G → gxg−1 ∈ G. Determine g,

in order φg be homomorphism.

6. Let φ : G → G′ be a homomorphism. Show that if |G| < ∞ then |φ(G)| <∞ and |φ(G)| | |G|.

7. Let φ : G → G′ be a homomorphism. Show that if |G′| < ∞ then |φ(G)| <∞ and |φ(G)| | |G′|.

8. Let φ : G → G′ be a homomorphism. Show that if |G| be a prime, then φ

must be a trivial homomorphism or monomorphism.


9. Let G, G′ and G” are groups. Show that if φ : G → G′ and γ : G′ → G”

be homomorphisms, then the composition γφ : G → G” be also homomor-

phism.

10. Let φ : G → G′ be homomorphism with Ker(φ) = H dan a ∈ G. Show

that {x ∈ G|φ(x) = φ(a)} = Ha.

11. Mention all isomorphisms that assign :

(a) Z2 ×Z3 into Z6

(b) Z2 ×Z5 into Z10

12. Determine the number of automorphisms in: Z2, Z6, Z8, Z, and Z12.

13. Let Γ be the set of groups. Defined a relation ∼ in Γ by: G ∼ G′ if G

isomorph to G′, ∀G, G′ ∈ Γ. Prove that ∼ is an equivalence relation.

14. LetG be a cyclic group with generator a. If φ : G → G′ is a isomorphism,

show that ∀x ∈ G, φ(x) can be expressed as the power of φ(a).

15. Let G be an abelian group. Show that if G′ isomorph to G, then G′ is also

abelian.

16. Let G be a cyclic group. Prove that cyclic is a structural properties of G.

17. Let [G, ·] be a group. An operation ∗ in G be defined as a ∗ b = b · a,

∀a, b ∈ G. Show that [G, ∗] is a group isomorph to [G, ·].

18. If G be a cyclic group of order n, then G isomorphs to Zn.Prove it.

19. Let G be a group and g ∈ G. Let αg(x) = gxg−1, ∀x ∈ G, show that αg is

an automorphism in G.

20. Let [S, ∗] be the group of real numbers except -1 under operation ∗ defined

by: a ∗ b = a + b + ab. Show that [S, ∗] isomorphs to [<∗, ·], where <∗ =

<− {0}.


21. Let [S, ∗] be the group of real numbers except -4. A mapping φ : R∗ → S

be defined by φ(x) = x− 4. Define a binary operation ∗ in S in order φ be

an isomorphism.

22. Let [S, ∗] be the group of real numbers except -t. A mapping φ : R∗ → S

be defined by φ(x) = x− t. Define a binary operation ∗ is S in order φ be

an isomorphism.

23. Let [S, ∗] be a group of real numbers. A mapping φ : <∗ → S be defined by

φ(x) = x3 + 1. Determine elements requirements in S and define a binary

operation ∗ in S in order φ be an isomorphism.

24. Let G be a group. Prove that permutations ρa : G → G, where ρa(x) = xa,

for a ∈ G and x ∈ G, forms a group G” that isomorphs to G.

Chapter 7

THE FACTOR GROUP

The factor group is a group that contains cosets of a subgroup. De-velopment of a factor group can be started by a homomorphism or anormal subgroup.

7.1 Factor Group Development by a Homomor-

phism

Theorem 7.1.1 Let φ : G → G′ is a homomorphism with Ker(φ) = H. Then

R/H = {a ∗H|a ∈ R} is a group with the binary operation:

(aH)(bH) = (ab)H

Dan the mapping µ : G/H → φ(G) defined by µ(a∗H) = φ(a), is an isomorphism.

Example 7.1.1 The mapping φ : Z → Zn defined by φ(m) = r, where r is the

reminder when m is divided by n, is a homomorphism. Since Ker(φ) = nZ,

hence Z/nZ is a group that isomorphs to Zn.

7.2 Factor Group Development by Normal Sub-

group

Theorem 7.2.1 Let H be a subgroup of group G. Multiplications of cosets of H,

that is defined by (aH)(bH) = (ab)H is well-defined if and only if aH = Ha,

44

VII. The Factor Group antonius cp 45

∀a ∈ G

Corollary 7.2.1 Let H be a normal subgroup of a group G. Then G/H =

{aH|a ∈ G} is a group with binary operation:

(aH)(bH) = (ab)H

Definition 7.2.1 The group G/H as described in the above corollary is Quo-

tient Group G modulo H

Example 7.2.1 If Z is an abelian group, then nZ is a normal subgroup of Z,

therefore it can be form a factor group Z/nZ without introducing a homomor-

phism.

Some conditions below are equivalent characteristics on a normal subgroup H

in a group G.

1. ghg−1 ∈ H, ∀g ∈ G and h ∈ H.

2. gHg−1 = H, ∀g ∈ G.

3. gH = Hg, ∀g ∈ G.

Show how that three conditions can be derived from the normal subgroup defin-

ition? To do this, use one of the characteristics above.

Definition 7.2.2 An isomorphism φ : G → G is called automorphism in G.

An automorphism Ig : G → G where Ig(x) = gxg−1 is called inner automor-

phism of G by g.

7.3 The Fundamental Homomorphism Theorem

Theorem 7.3.1 Let H be a factor group of a group G. Then φ : G → G/H

defined by φ(a) = aH, is a homomorphism with Ker(φ) = H.


Theorem 7.3.2 The Fundamental Homomorphism Theorem. Let φ :

G → G′ be a homomorphism with Ker(φ) = H. Then φ(G) is a group, and

the mapping µ : G/H → φ(G), defined by µ(aH) = φ(a), is an isomorphism. If

γ : G → G/H is a homomorphism that is defined by γ(a) = aH, then ∀a ∈ G,

φ(a) = µγ(a).

7.4 Exercises on the Factor Group

1. Compute the order of each factor group below.

(a) Z6/ < 3 >

(b) (Z4 ×Z2)/ < (2, 1) >

(c) (Z2 ×Z4)/ < (1, 1) >

(d) (Z2 × S3)/ < (1, ρ1) >

(e) (Z4 ×Z12)/(< 2 > × < 2 >)

(f) (Z3 ×Z5)/({0} × Z5)

(g) (Z12 ×Z18)/ < (4, 3) >

(h) (Z11 ×Z15)/ < (1, 1) >

2. Compute the order of each element of the following factor group.

(a) 5+ < 4 > in Z12/ < 4 >.

(b) 26+ < 12 > in Z60/ < 12 >.

(c) (2, 1)+ < (1, 1) > in (Z3 ×Z6)/ < (1, 1) >.

(d) (3, 1)+ < (1, 1) > in (Z4 ×Z4)/ < (1, 1) >.

(e) (3, 1)+ < (0, 2) > in (Z4 ×Z8)/ < (0, 2) >.

(f) (3, 3)+ < (1, 2) > in (Z4 ×Z8)/ < (1, 2) >.

(g) (2, 0)+ < (4, 4) > in (Z6 ×Z8)/ < (4, 4) >.

3. Determine true or false each statement below and give the reason.


(a) A factor group G/N can be developed if and only if N is a normal

subgroup of G.

(b) Every subgroup of an abelian group is normal.

(c) An automorphism inner in an abelian group must be an identity func-

tion.

(d) Every factor group of a finite group is also finite.

(e) Every factor group of an abelian group is also abelian.

(f) Every factor group of a non abelian group is also non abelian.

(g) Z/nZ is a cyclic group of ordern.

4. Let H be a normal subgroup of a group G, and let m = (G : H). Show

that am ∈ H, for every a ∈ G!

5. Show that intersection of normal subgroups is also a normal subgroup!

6. Show that if a finite group G has exactly one subgroup H of certain order,

then H is also normal subgroup of G!

7. Show that if H and N are subgroup of group G and N is a normal subgroup,

then H ∩ N is normal in H! Show using an example that H ∩ N is not

necessary a normal of G!

Bibliography

[1] Frank Ayres, JR, 1965, Theory and Problems of Modern Algebra, Schaum

Publishing Co, New York.

[2] Gede Kerta Widarma, 1990, Pengantar Teori Group (Struktur Aljabar I),

Fakultas Keguruan dan Ilmu Pendidikan, Universitas Jember.

[3] I.N. Herstein, 1975, Topics in Algebra, John Wiley & Sons, New York.

[4] Jimmie Gilbert, Linda Gilbert, 1992, Elements of Modern Algebra, PWS-

KENT Publishing Co, Boston.

[5] John B. Fraleigh, 1989, A First Course In Abstract Algebra, Addison-Wesley

Publishing Company, Canada.

[6] M.D. Raisinghania, R.S. Aggarwal, 1980, Modern Algeba, S. Chand & Com-

pany, New Delhi.

[7] Theresia M.H. Tirta Seputro, 1992, Pengantar Dasar Matematika Logika

dan Teori Himpunan, Penerbit Erlangga, Jakarta.

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INTRODUCTION TO THE GROUP€¦ · INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: [email protected] Mathematics Education Study Program