INTRODUCTION TO THE GROUP THEORY

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Lecture Notes on Structure of Algebra

INTRODUCTION TO THE GROUP

THEORY

By :

Drs. Antonius Cahya Prihandoko, M.App.Sc

e-mail: antoniuscp.fkip@unej.ac.id

Mathematics Education Study Program

Faculty of Teacher Training and Education

The University of Jember

2010

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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 groupandhomomorphism. The descrip-tion is started with the concepts ofsetandfunction, that are basic of all concepts in this book. After getting a good knowledge in the concepts ofsetand 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 provideshomomorphism,isomorphismandCayley theorem. Finally, the last chapter provides the concept of factor group.

Jember, August 2010 Antonius C. Prihandoko

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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 of Algebra. As a group is basically a set and a homomorphism is a func-tion, it is important to describe set theory and function first before discussing the main contents of this subject. The aimof this chapter is that the students have understanding on set, function, partition, equivalence relation and binary operation. Outcome of this chapter is 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 ofgroup. Ifgroupis identified as ”unity of things”, then there will be a question about definition ofunity. 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.

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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 calledempty 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 calledwell-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 ofA 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 ifA_⊆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

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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 i₆=j then Ai∩Aj =φ;

2. _∪n

i=1Ai =A

Example 1.2.2 1. The integers set, _Z, can be partitioned to set of odd inte-gers and set of even inteinte-gers.

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, _ℜ, 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:

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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 aand bbe two integers on_Z andn 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

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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 thatφis surjective, it has to be proved that for every b_∈B, there exist a_∈A such thatφ(a) =b.

1.4

Binary Operation

Definition 1.4.1 Binary operationon 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=cthen

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 (_ℜ), on the set (_Z), on the set (_C), or on the set (_Q) are binary operation.

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3. The usual addition is not a binary operation on the set_ℜ∗

=_ℜ−{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. _∃!csuch 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 (_ℜ), on the set (_Z), on the set (_C), or on the set (_Q) are commutative and associative.

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

multiplication onM2,2(ℜ)is associative but not commutative. Please inspect

this!

3. Substraction on _ℜ 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, thenK =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

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(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_{∈ ℜ|}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 _ℜif x_≥y

(c) x_∼y on the set _ℜif _|x_|=_|y_|

(d) x_∼y on the set _ℜif _|x₋y_{| ≤}3 (e) x_∼y on the set _Z+

if xand y have the same digits. (f) x_∼y on the set _Z+

if xand 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 setS 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) = sinx, f4(x) =

x5

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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 bya_∗b =a₋b

(b) _∗defined on _Q bya_∗b =ab+ 1 (c) _∗defined on _Q bya_∗b = ab

2 (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_∗ bya_∗b =a₋b. (b) On the set _Z+

, define_∗ bya_∗b =ab_.

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(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 _∗bya_∗b=c, where cis 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 setS, 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 existsa, b_∈S, such thata_∗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 setS 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 ordered pair of elements of S.

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

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.

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(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)_∗′

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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 Abstract Algebra). It contains the definition, properties, order, and concepts of group 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 operationdefined on the set. A system that consists of a non empty setGand a binary operation_∗defined on the set is calledgrupoid. If the binary operation on the grupoid is associative, then the system is called bysemi group. If the semi group consists of identity element,e, such that for every a_∈Gsatisfy

a_∗e=e_∗a=a, then the system is calledmonoid. And if each element of monoid has own inverse, that is for eacha _∈G, _∃a−1

∈Gsuch thata_∗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).

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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 ofGhas 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 _ℜ 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,_∗>iscommutativeif(_∀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 aree′

and eand 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 thene′

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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

andc, where both of them are inverse ofa. So that (b_∗a)_∗c=cdanb_∗(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−1

n ∗a −1

n−1 ∗ · · · ∗a

−1 1

.

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−1

b.

Theorem 2.2.7 If an empty set G under binary operation _∗ satisfy the axioms: closed, associatif, and the equationa_∗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

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Definition 2.3.1 The result of operation of m factors, a_∗a_∗a_∗a_{∗ · · · ∗}a is represented byam_{; The result of operation of}_m_factors,_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(_ℜ,_×), 23

= 2_×2_×2 = 8; 2−1 = 1

2 so that2

−3 = 1 2× 1 2× 1 2 = 1 8; 20

= 1, since 1is 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 Gis the number of elements of

G. If the number of elements of Gis infinite, then the order of Gis 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 thatan ₌_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, since5is 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

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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 ofa is infinite then all power of a are distinct, that is if r₆=s thanar₆₌_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 andH 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.

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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,_∗>, thenH_∗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 andK 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 is21 = 2; 22

= 4; 23

= 3; and 24

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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(_ℜ) under addition matrix. (k) The set M2(_ℜ) under multiplication matrix.

(l) The set of all integers divisible by 5, under addition. (m) The set of all vectors in _ℜ2

of the form (x,3x) under addition vector. (n) The set of all vectors in _ℜ2

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) = 1

z, 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.

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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 Gsuch that a_∗a=e!

6. Let_∗be a binary operation on a setS, an elementx_∈Sis calledidempotent

for _∗ if x_∗x=x. Show that a group has exactly one idempotent element. 7. If Gis 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, thena_∗b=b_∗a! 9. LetG 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

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(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 ifGis an abelian group with identity elemente, then all element

x_∈G satisfying the equation x2

=e forms a subgroup H of G!

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

ax_} is a subgroup ofG!

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, thenH_∩K

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Chapter 3

CYCLIC GROUP

As discussed before that the set of all power of an element of a group forms a subgroup. It derives the concept of a group containing all power 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).

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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 ifb 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 n_Z 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 Gis 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 ofr ands 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

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3.2

Subgroup of Finite Cyclic Group

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

1. If Gis 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 Letn be a fixed positive integer and lethandk be any integers. The remainder r whenh+k is divided byn 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 n

d 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 12

4 = 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!

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

4. Let +n be notation of addition modulo n. Compute: 13 +178, 21 +3019,

26 +4216, and 39 +5417.

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) Qunder 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.

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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 ordern containing an element of ordern, prove that the group is cyclic.

14. In a cyclic group of ordern, show that there exists element of orderk, where

k is a factor ofn.

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, wherer is an integer ≥1.

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

exactlym solutions in Gfor each positive integer m that divides n. 19. Show that_Zp has no proper subgroup if p is a prime number.

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Chapter 4

GROUP OF PERMUTATIONS

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

4.1

Permutation

Definition 4.1.1 Permutationin a setA is a one to one function fromAonto

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





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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 LetA=_{1,2,3_} thenS3 has3! = 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)

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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 µhas 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

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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 permutationσ 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 τ σ 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.

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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}.

2. Determine true or false the following statements and give the reason. (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 eachn.

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

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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





7. Determine true or false the following statements and give the reason. (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. LetGbe a group and abe a fixed element inG. Show that mapλa:G−→

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Chapter 5

COSET AND THE LAGRANGE

THEOREM

A subgroup of a group has cosets. The set of the cosets determines a partition on the group. Furthermore, there is a one to one correspon-dence between the subgroup and its coset. This concept leads the relationship 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_∼Lb ⇔a −1

b_∈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−1

x _∈ H_}, or equivalently equal to _{x _∈ G_|a−1

x = 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.

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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 calledleft 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 _Zis a group under integer addition and3_Z=_{{· · ·} ,₋6,₋3,0,3,6,9,_{· · · }}

is a subgroup of _Z. The left cosets of 3_Z in _Z are

• 3_Z=_{{· · ·} ,₋6,₋3,0,3,6,9,_{· · · }}

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

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

It can be shown that _{3_Z, 1 + 3_Z, 2 + 3_Z} 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 Gbe a group and H be a subgroup ofG, 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.

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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 IfG be a prime ordered group thenG 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) 4_Z of group_Z.

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(e) <18>of group _Z36. (f) _{ρ0, ρ2} of groupD4. (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.

3. Determine true or false the following statements and give the reason. (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−1

hg _∈ H, _∀g _∈ G and

∀h_∈H. Show that each left coset of H, gH, same as the right coset, Hg! 5. LetH be a subgroup of a group G. Show that if the partition on Gby the

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

g−1

hg_∈H is satisfied, _∀g _∈G and _∀h_∈H!

6. Give the proof if the following is true, or give acounter-exampleif 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 a2

H =b2

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7. If G be a group of order pq, where both of p and q are prime, show that each proper subgroup of Gis 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 groupG, 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 ifa =hbk, for anyh_∈H dan k _∈K. (a) Prove that _∼ is an equivalence relation in G!

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Chapter 6

GROUP HOMOMORPHISM

This chapter is the beginning of second part of the subject of Algebra Structure. In the first part, we discuss about group and its properties. In this second part we will discuss about a kind of function that assigns a group to another group. This chapter provides the definition of homomorphism 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 Gand G′

.

Notes : For simplify the notation we will not write the binary operation symbol. LetGbe a group anda, b_∈Gthen binary operation onaand bbe written asab.

Between any two groupsGandG′

, there is at least a homomorphismφ:G_→ G′

called trivial homomorphism defined by φ(g) =e′

for eachg _∈G where e′

is the identity element of G′

.

Example 6.1.1 Let α:_{Z → Z}n defined by α(m) =r, where r is the remainder

when m is divided byn. Then α is a homomorphism.

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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 φbe a mapping of a setX into a setY, and letA_⊆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′

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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 isomorphismφ: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 !

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2. Show thatφ is one-to-one function. 3. Show thatφ is onto.

4. Show thatφ(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;

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

17 _{∈ Z} but 17 _6∈ 3_Z. This properties is non structural properties. In the fact _Z and 3_Z are isomorph since the function φ : _{Z → ∋Z}, where

φ(n) = 3n, is a isomorphism.

2. It can not be said that _Z and _Q are not isomorph since 1

2 ∈ Q but 1 2 6∈ Z.

But it can be said that _Z and _Qare not isomorph since _Z is cyclic, but _Q is 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 groupG, collect all permutation in Ginto 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 Gto 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 → ℜ} defined by φ(x) =x. (b) φ:_{ℜ → Z} defined by φ(x) =_⌊x_⌋.

(c) φ:_Z6 → Z2 defined byφ(x) = the remainder when xis divided by 2. (d) φ:_Z9 → Z2 defined byφ(x) = the remainder when xis 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.

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(h) φ: [_ℜ,+]_→[_ℜ,_·] defined byφ(x) = 2x

(i) φi :Gi →G1×G2×...×Gi×...×Grdefined byφi(gi) = (e1, e2, ..., gi, ..., er),

dimanagi ∈Gi dan ej elemen identitas dalam Gj.

(j) φ:G_→Gdefined 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_ℜℜ

that are differentiable on all order. (l) φ:F _→Rdefined byφ(f) = R4

0 f(x)dx, whereF is the addition group of all continue function_ℜℜ

; _ℜis the addition group of real numbers. (m) φ:F _→F defined by φ(f) = 3f, where F is the addition group of all

function_ℜℜ

.

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

of all matrices 2x2, and_ℜ is the addition group of all real numbers. 2. Let F = _ℜℜ

be an addition group, D = _{f _{∈ ℜ}ℜ

|f dif f erentiable _}. Is

φ:f _∈D_→f′

∈F homomorphism or not? If yes, find Ker(φ).

3. LetGbe a group. Ifφ:_{Z × Z → G} is a homomorphism and letφ(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. Letφ :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

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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. LetG be an abelian group. Show that ifG′

isomorph to G, then G′

is also abelian.

16. LetG 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 Gbe a cyclic group of order n, then G isomorphs toZn.Prove it.

19. Let Gbe a group and g _∈G. Let αg(x) =gxg−1, ∀x∈ G, show thatαg 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 [_ℜ∗

,_·], where _ℜ∗

=

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

be defined byφ(x) = x₋4. Define a binary operation_∗ inS 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 orderφ be an isomorphism.

23. Let [S,_∗] be a group of real numbers. A mapping φ:_ℜ∗

→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. LetGbe a group. Prove that permutationsρa :G→G, where ρa(x) = xa,

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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 a normal 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 mappingµ:G/H _→φ(G)defined byµ(a_∗H) = φ(a), is an isomorphism.

Example 7.1.1 The mapping φ : _{Z → Z}n defined by φ(m) = r, where r is the

reminder when m is divided by n, is a homomorphism. Since Ker(φ) = n_Z, hence _Z/n_Z is a group that isomorphs to _Zn.

7.2

Factor Group Development by Normal

Sub-group

Theorem 7.2.1 LetH be a subgroup of groupG. Multiplications of cosets ofH, that is defined by (aH)(bH) = (ab)H is well-defined if and only if aH = Ha,

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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 _∈Gand 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

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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)>.

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(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.

INTRODUCTION TO THE GROUP THEORY

Lecture Notes on Structure of Algebra

Table of Contents

1 SET AND FUNCTION 1

Chapter 1

Partition and Equivalence Relation

Function

Binary Operation

Exercise on Set and Function

Chapter 2

Properties of Group

Order of Group and Element

Subgroup

The Exercise for Concepts of Group

Chapter 3

Subgroup of Finite Cyclic Group

Exercises on Cyclic Group

Chapter 4

Orbit and Cycle

Exercises on Group of Permutation

Chapter 5

The Lagrange Theorem

Exercises on Coset and the Lagrange

Chapter 6

Isomorphism dan Cayley Theorem

Exercises on group Homomorphism

Chapter 7

Factor Group Development by Normal

The Fundamental Homomorphism Theorem