phanny's report paper-mathematics-rupp

saklviTüal½yPUminÞPñMeBj ROYAL UNIVERSITY OF PHNOM PENH

THE MULTIPLICATIVE GROUP OF NON-ZERO COMPLEX NUMBERS IS ISOMORPHIC TO

THE MULTIPLICATIVE GROUP OF THE UNIT CIRCLE

RkumplKuNéncMnYnkuMpøicminsUnüGuIsUm:kPIk eTAnwgRkumplKuNénrgVg;Ékta

A Thesis In Partial Fulfillment of the Requirement for the Degree of

Master of Science (Mathematics)

niekçbT dak;CUnkñúgcMENkmYysMrab;karbMeBjtMrUvkarénkarbBa©b;

briBaaØbRtCan;x<s;viTüasaRsþ ¬EpñkKNitviTüa¦

Guit panI

ITH Phanny

tula 2009 October 2009

Royal University of Phnom Penh

TO WHOM IT MAY CONCERN

Name of program: Mathematics

Name of candidate: ITH Phanny

Title of thesis: “The Multiplicative Group of Nonzero Complex Numbers is Isomorphic to the Multiplicative Group of the Unit Circle.”

This is to certify that the research carried out for the above titled master’s thesis was completed by the above named candidate under my direct supervision. This thesis material has not been used for any other degree. I played the following part in the preparation of the thesis:

I suggested the problem, supervised and guided the research, made comments on drafts of the thesis, and ensured that the finished product represented the candidate’s own work and met standards for a Masters degree.

Supervisor(s) (Sign)………………………………

Date: July 06, 2009


TO WHOM IT MAY CONCERN

This is to certify that the thesis that I am ITH Phanny

hereby present entitled “THE MULTIPLICATIVE GROUP OF NON-ZERO COMPLEX NUMBERS IS ISOMORPHIC TO THE MULTIPLICATIVE GROUP OF THE UNIT CIRCLE.”

for the degree of Master of Science at the Royal University of Phnom Penh is entirely my own work and, furthermore, that it has not been used to fulfill the requirements of any other qualification in whole or in part, at this or any other University or equivalent institution.

No reference to, or quotation from, this document may be made without the written approval of the author.

Signed by (the candidate): …………………..

Date: October 03, 2009

Countersigned by the Chief Supervisor

………………………………………

Date:………………………………..


Master of Science in Mathematics

Contents

CONTENTS

ABSTRACT .......................................................................................................... ii

ACKNOWLEDGEMENT ....................................................................................... iii-iv

1. INTRODUCTION

1.1 DEFINITION 1.1 (OPERATION) ........................................................ 2

1.2 DEFINITION 1.2 (GROUPS) .............................................................. 2

1.3 DEFINITION 1.3 (MULTIPLICATIVE GROUP) ................................... 2

1.4 DEFINITION 1.4 (ABELIAN GROUP) ................................................. 4

1.5 DEFINITION 1.5 (CARDINALITY) ...................................................... 5

1.6 DEFINITION 1.6 (SUBGROUP) .......................................................... 7

1.7 DEFINITION 1.8 (CYCLIC GROUP) .................................................... 9

1.8 DEFINITION 1.9 (GROUP HOMOMORPHISM) ...................................... 10

1.9 DEFINITION 1.10 (KERNEL OF HOMOMORPHISM) ............................ 11

1.10 DEFINITION 1.11 (GROUP OF UNIT CIRCLE) .................................. 12

2. OTHER DEFINITIONS, MAIN COROLLARY AND THEOREMS

2.1 THEOREM 2.1 (HOMOMORPHISM THEOREM) .................................. 14

2.2 DEFINITION 2.2 (LEFT AND RIGHT COSETS) ..................................... 14

2.3 DEFINITION 2.4 (NORMAL SUBGROUP) ........................................... 16

2.4 DEFINITION 2.7 (QUOTIENT GROUP) ................................................ 17

2.5 THEOREM 2.9 (ISOMORPHISM GROUP) ............................................ 18

2.6 MAIN THEOREM 2.10 ( ffG Imker/ ≅ ) ........................................ 19

2.7 MAIN COROLLARY 2.11 ( 'ker/ GfG ≅ ) ....................................... 20

2.8 MAIN THEOREM 2.13 ( :group isomorphim f∃ → ) ............. 21

2.9 MAIN THEOREM 2.14 ( * 1≅ S ) ..................................................... 22

3. BIBLIOGRAPHY

3.1 HISTORY OF GROUP DEVELOPMENT ............................................... 24

3.2 REFERENCES ................................................................................... 28



Contents

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ABSTRACT

Nowadays, everyone agrees that some knowledge of linear algebra, groups, and

commutative rings is necessary, and these topics are introduced in undergraduate courses.

We have continued to study our great deals of mathematics that is why the research

report given me the opportunity to do further comprehension in it. This study is to consider

that real one of the importance which is called group (isomorphism). The main purpose of

this paper is to show that “the multiplicative group of non-zero complex numbers is

isomorphic to the multiplicative group of the unit circle” as well as consisting of definitions,

examples, some basic group theory, and the relations, especially, subgroup and quotient

group needed for the proof. To show that two groups are isomorphic usually means that we

need to construct a precise isomorphism. In addiction, we want to understand the relation

between group and group isomorphism (i.e., homomorphism and bijective).

As a final point, we have to understand what the multiplicative group of non-zero

complex numbers and the multiplicative group of the unit circle is so we will be able to

prove that they are isomorphic.



Contents

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ACKNOWLEDGEMENTS

First I would like to thank my grandparents and my parents for everything long

eternal love and support with my study. I would like to thank my advisor, Professor Will

Murray, Ph.D. and other professors who come to give lectures in Cambodia such as from

France, Sweden, the USA, and from Japan.

Next I would like to thank my lecturers and professors at the Royal University of

Phnom Penh (RUPP) for their varieties of advice and motivation.

Finally, I would like to thank all my teachers and classmates for their help and

encouragements.



Contents

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To my grandparents, parents, brothers and sisters



Introduction

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1. INTRODUCTION

When is a group isomorphic to another group? What is the

multiplicative group of non-zero complex numbers? And what is the

multiplicative group of all complex numbers with modulus 1? As we

will see we need some properties and concrete definitions as

subsequences.

DEFINITION 1.1: A binary operation on a set G is a function

:f G G G× → such that zyxyx =∗),( . On normal binary operations are

addition, multiplication, and subtraction, respectively, the functions

yxyx +),( , xyyx ),( , yxyx −),( . Division of integers is not a

binary operation on integer (because an integer divided by an integer

need not to be an integer).

DEFINITION 1.2: G is a set equipped with the binary operation ∗ is

a group if it satisfies:

(i). The associative law holds: for every , , : ( * )* *( * )x y z G x y z x y z∈ = ,

(ii). There exists an element Ge∈ , called the identity, with

exxxe ** == for all Gx∈

(iii). Any Gx∈ has an inverse; there is any ' ,x G∈ with xxexx '*'* == .

DEFINITION 1.3: A multiplicative group is a group whose operation

is identified with multiplication. As with ordinary multiplication, the

multiplication operation on group elements is either denoted by a dot or

omitted entirely ( )x y or xy⋅ . In a multiplicative group, the identity

(unit) element is denoted by 1 and the inverse of the element x is

written as 1x− .

Example 1: The set of integers under ordinary multiplication is

not a group. Since Property (iii) does not hold. For example, there is no

integer b such that 15 =b .



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Example 2: The set + of the positive rational numbers is a

group under ordinary multiplication. The inverse of a is a1 .

Example 3: The set of integers under subtraction is not a group,

since the operation is not associative.

Example 4: The set ∗ of non-zero real numbers is a group under

ordinary multiplication. The identity is 1 and the inverse of a is 1−a .

Example 5: The set {0, 1, 2, 3} is not a group under multiplication

modulo 4 . Although 1 and 3 have the inverse, the element 0 and 2 do

not.

Example 6: The solutions of the polynomial 014 =−x , i.e., the

subset {1, 1, , }i i− − of the complex numbers is a group under

multiplication. Note that 1− is inverse of 1, whereas the inverse of i− is

i .

Following this example, we are going to recall the complex

numbers. The set of all complex numbers is usually denoted by . The

set {1, }i will form a basis for the extension field over the real numbers

field, which means every element of the extension field can be written

as iba ⋅+⋅1 (where ,a b∈ and 12 −=i ). Equivalently, the complex

number bia + can be written as the point ( ) 2,a b ∈ in the complex plane

coordinatized by a horizontal real axis and a vertical i (or imaginary)

axis. To see this when each real number a is identified with the

complex number ia ⋅+ 0 , then the real numbers field, becomes the

complex numbers field, .

The absolute value (or modulus or magnitude) of a complex

number irez θ= is defined by rz = . Algebraically, if biaz += , then

22 baz += is the distance from z to the origin. Note that if z and w

are complex numbers with polar coordinates ),( αr and ),( θs ,

respectively, then the polar coordinates of zw are ( , )rs α θ+ and so



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wzzw = . This formula is correct if πθα 2≤+ ; otherwise, the angle

should be πθα 2−+ . For example, when we want to add 270150 + the

answer is 420 , then we adjust the answer 60360420 =− . In particular,

if wz ==1 , then 1=zw ; that is, the product of two complex numbers

on the unit circle also lies on the unit circle.

On the other hand, thinking in terms of the ordinary addition

where only numbers between 0 and 1 are allowed we need to throw

away digits before the decimal point. For instance, when we add

0.784+0.925+0.446, the answer is 2.155, but we throw away the

leading 2, so the answer is 0.155 (in the unit circle). Thus, the four

complex number zeros of 14 =x (the fourth roots of unity) are located at

points around the circle of radius 1 or the unit circle (see Example 6).

Note also that we refer to the group ( , )G + or simply to the group G .

Example 7: The set * of all non-zero complex numbers forms a

group under multiplication since:

1. Associativity:

For any *( ), ( ), ( )a ib c id e if+ + + ∈ where , , , , ,a b c d e f ∈ implies

[ ] [ ][ ] [ ]

[ ] [ ][ ][ ]

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )

( )( ) ( )

a ib c id e if a ib ce df i cf de

ce df a cf de b i cf de a ce df bace adf bcf bde i acf ade bce bdfac bd e ad bc f i ac bd f ad bc e

ac bd i ad bc e if

a ib c id e if

+ + + = + − + +

= − − + + + + −

= − − − + + + −

= − − + + − + +

= − + + +

= + + +

2. Identity: the identity is 1; that is, *1 (1 0)i= + ∈ .

3. Inverse: iba

bba

aiba 22221)(

+−

++

=+ − is the inverse.

Since ( )2 2

2 2 2 2 2 2 2 2 2 2 2 2 1a b a b ab aba ib i ia b a b a b a b a b a b

⎛ ⎞− −⎛ ⎞ ⎛ ⎞+ + = + + + =⎜ ⎟⎜ ⎟ ⎜ ⎟+ + + + + +⎝ ⎠ ⎝ ⎠⎝ ⎠.

Thus, * is a group.



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Example 8: The set 1 { : 1}z z= ∈ =S , is the group whose

operation is multiplication of complex numbers with modulus 1 since:

1. Associativity:

We have ( ) ( ) 1z w v z w v= = for any 1, ,z w v ∈S ; that is, the

complex multiplication is associative.

2. Identity: the identity is 1 (which has modulus 1); that is,

zzz == 11 and

3. Inverse: if 1=z , then 1=z ; that is, the inverse of any

complex numbers of modulus 1 is its complex conjugate,

which also has modulus 1.

Therefore, 1S is a group.

DEFINITION 1.4: If the binary operation ∗ is commutative; that is,

xyyx ** = , then G is called commutative or abelian group.

Example 9: The groups ( , ), ( , ), ( , ), ( , ), ( / , )n+ + + + + are

abelian groups.

DEFINITION 1.5: The cardinality (number of elements) of a group

G (finite) is called the order of G , denoted by ( )o G . Let x be an

element of the group G . If there exists a positive integer n such that

exn = (in addition notation, this would be 0=nx ), then x is said to have

finite order and the smallest such integer is called the order of x ,

denoted by ( )o x . If no such integer exists, then one says that x has

infinite order.

Example 10: The group of integers under addition has infinite

order since 3 3 3 3+ + + ⋅⋅⋅ + is never 0 ; whereas the group (10) {1, 3, 7, 9}U =

under multiplication modulo 10 has order 4.

LEMMA I: Let G be a group.

(i). The cancellation law holds; that is, if either bxax ** = or

xbxa ** = , then ba = .



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(ii). The identity element Ge∈ is unique; that is, exxxe ** == for

all Gx∈ .

(iii). Any Gx∈ has a unique inverse; there is only one Gx ∈' , with

xxexx '*'* == (this 'x will be denoted by 1−x ).

(iv). xx =−− 11)( for any Gx∈ .

Proof of LEMMA I

(i). Choose 'x with xxexx '*'* == , then

bbxxbxxaxxaxxaea ====== *)'*()*'*()*'*(*)'*(* .

Similarly, .)'*(*'*)*('*)*()'*(** bxxbxxbxxaxxaeaa ======

(ii). Let Ge ∈0 such that 00 ** exxxe == for all Gx∈ . Setting ex = ,

then the second equation becomes 0*eee = . On the other hand, by

definition of e gives 00* eee = . Therefore, 0ee = .

(iii). Assume that Gx ∈" satisfies xxexx "*"* == . We have the

equation '* xxe = and multiply this equation on the left by "x , then

''*'*)"*()'*"*("*" xxexxxxxxexx ===== .

(iv). By definition, 111111 )(**)( −−−−−− == xxexx .

Therefore, xx =−− 11)( , by using (iii).

From now on, we will denote the product yx * by xy and the

identity by 1 instead of e . When a group is abelian, we will use the

addition notation yx + , in this case we will denote the identity by 0 and

the inverse of an element x by x− instead of 1−x .

Example 11: 0 is the identity of the addition groups , , ,

and 1 is the identity of the multiplication groups , , , .



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Example 12: The addition groups , , have the inverses x−

for any , ,x ∈ .

DEFINITION 1.6: G is a group and H is a nonempty set of G and is

stable in G on the operation ∗ . One says that H is a subgroup of G if

H is a group. That is,

(i). For any Hyx ∈, , then x y H∗ ∈ .

(ii). If Ge∈ , then He∈ .

(iii). If Hx∈ , then Hx ∈−1 .

Note that if we want to say that H is a subgroup of G , but not

equal to G itself, denoted by GH < . Such a subgroup is called a proper

subgroup. Observe that }{e and G are always subgroups of G , where

}{e denote the subset consisting of the single element e . The subgroup

that is not }{e called a nontrivial subgroup of G .

Example 13: The addition rational group, ( , )+ is a subgroup of

the addition real group, ( , )+ . The multiplicative non-zero rational

group, *( , )⋅ is a subgroup of the multiplicative non-zero real, *( , )⋅ .

The multiplicative non-zero real, *( , )⋅ is a subgroup of the

multiplicative non-zero complex numbers, *( , )⋅ .

Example 14: Let M be the following subset of 2 ( )GL :

: , 2 , 0a b

M a d c b ad bcc d

⎧ ⎫⎡ ⎤= = = − − ≠⎨ ⎬⎢ ⎥

⎣ ⎦⎩ ⎭. Show that M is a subgroup of

2 ( )GL .

~ Since 2: , 2 , 0 : 2 02

a b a bM a d c b ad bc ad b

c d b a⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤

= = = − − ≠ = + ≠⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭,

then for any 1 1 2 2

1 1 2 2

,2 2a b a b

Mb a b a

⎡ ⎤ ⎡ ⎤∈⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

;

Mbbaaabba

abbabbaaaabbbaab

abbabbaaabba

abba

∈⎥⎦

⎤⎢⎣

⎡−+−+−

=⎥⎦

⎤⎢⎣

⎡+−−−+−

=⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡− 21212121

21212121

21212121

21212121

22

22

11

11

2)(22

2222

22 .



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~ The identity 2 2 2

1 0( )

0 1E GL E M

⎡ ⎤= ∈ ⇒ ∈⎢ ⎥⎣ ⎦

.

~ Suppose ⎥⎦

⎤⎢⎣

⎡−

=11

111 2 ab

baA

1 12 2 2 211 1 1 11 1 1 11

1 2 21 1 1 11 1 1 1

2 2 2 21 1 1 1

2 212 22

22 2

a ba b a ba b a b

A Mb a b aa b b a

a b a b

−

−

−⎡ ⎤⎢ ⎥+ +−⎡ ⎤ ⎡ ⎤ ⎢ ⎥⇒ = = = ∈⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎛ ⎞+ −⎣ ⎦ ⎣ ⎦ −⎢ ⎥⎜ ⎟+ +⎢ ⎥⎝ ⎠⎣ ⎦

Thus, M is a subgroup of 2 ( )GL .

THEOREM 1.7: Let G be a group and H be a nonempty subset of G .

Then H is a subgroup of G if and only if φ≠H and 1* −yx is in H

whenever x and y in H .

Proof

)(⇒ Suppose H is a subgroup of G , then He∈ by definition 1.6,

so that φ≠H . On the other hand, if Hy∈ , then Hy ∈−1 . Therefore, for

any HyxHyx ∈∈ −1*,, .

)(⇐ Suppose that φ≠H , then there exists at least one element

Hx∈ such that Hexx ∈=−1* . Moreover, for any Hy∈ , then

Hyey ∈= −− 11 * . Finally, for any HyxyxHyx ∈=∈ −− 11)(**,, .

Note that if the operation in G is addition, then for any

HyxHyx ∈−⇒∈, .

Example 15: Let G be an abelian group under multiplication and

let n be a fixed positive integer. Then }:{ GxsomeforbxGbH n ∈=∈= is

a subgroup of G .

Since ... n

n times

e e e e e H⋅ ⋅ ⋅ = = ∈ , then φ≠H .

Next we suppose that the elements 1b and 2b are in H . Then there must

exist 21,, bybxwithGyx nn ==∈ so that 1121 )( −− = nn yxbb . Since G is



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abelian, we may write 1121 )( −− = nn yxbb as nnn xyyxbb )()( 111

21−−− == which is

the right form of H . Thus, H is a subgroup of G .

Example 16: Let G be an abelian group under multiplication

with identity e . Then }:{ 2 GxxH ∈= is a subgroup of G (being said

that H is the set of all the “squares”).

Since Heeee ∈== 2 , then φ≠H . Next we want two elements of

H in H , saying 2a and 2b . We must show that 122 )( −ba also has the

correct form; that is, 122 )( −ba is the square of some element. Since G is

abelian, we may write 122 )( −ba as 21)( −ab which is the correct form.

Therefore, H is a subgroup of G .

Example 17: The multiplicative group of all complex numbers

of absolute value 1, 1 { : 1}z z= ∈ =S forms a subgroup of * , the

multiplicative group of all non-zero complex numbers.

DEFINITION 1.8: Let G be a group and 'G be a subgroup of G . One

says that 'G is a cyclic subgroup of G generated by x if there is 'Gx∈

such that ' { : }nG x x n= < > = ∈ . We will call a group that consists of a

single element Gx∈ (called the group generator) a cyclic group

generated by x and denoted by >< x if written in multiplication

notation as 2 1{1, , ,..........., }nG x x x x −= < > = where 1=nx . In particular,

observe that the exponents of x include all negative integer as well as

0 and the positive integers ( 0x is defined to be the identity). A finite

cyclic group generated by x is necessarily abelian, and can be written

in addition notation as { : } {0, , 2 ,........., ( 1) }G x nx n x x n x= < > = ∈ = − .

Clearly if there is n such that 0=nx , we say that >< x is finite.

Otherwise x is infinite. Similarly, if G is an infinite cyclic group

generated by x , then G must be abelian and can be written as 2{1, , ,...............}x x or in addition notation {0, , 2 ,.................}x x . In this case,

G is isomorphic to additive group of all integers.



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Example 18: The ( , )+ is a cyclic group generated by 1 or 1− ;

that is, 1 { (1) : }n n= < > = ∈ and 1 { ( 1) : }n n= < − > = − ∈ .

Example 19: The set )},1,1({ ⋅− is a cyclic multiplicative group of

order 2 where the generator is 1− . Check 1 { ( 1) : } { 1,1}n n< − > = − ∈ = − .

Example 20: The set { : 0, 1, 2, , 1}knC k nω= = − of all complex

thn roots of unity is a cyclic multiplicative group of order n where the

generator 2 2 2cos sin

ine i

n n

π π πω ⎛ ⎞ ⎛ ⎞= = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

is any thn roots of unity.

Example 21: Recall Example 10, }9,7,3,1{)10( =U ,

3 {1,3,7,9}< > = . Check that: 1 2 3 4 5 4 6 5 13 3, 3 9, 3 7, 3 1, 3 3 3 3, 3 3 3 9, , 3 7−= = = = = ⋅ = = ⋅ = = (since 173 =⋅ ),

2 1 1 3 2 1 4 5 4 1 6 5 13 3 3 9, 3 3 3 3, 3 1, 3 3 3 7, 3 3 3 9,− − − − − − − − − − − − −= ⋅ = = ⋅ = = = ⋅ = = ⋅ =

DEFINITION 1.9: If HGf →: , where G and H are groups, then f is

said to be a group homomorphism if for any , ,x y G∈ we have

)()()( yfxfxyf = . If GGf →: is a group homomorphism, then f is

called endomorphism.

A homomorphism f from a group G to a group H and injective

is called monomorphism. A homomorphism f from a group G to a

group H and surjective is called epimorphism. A homomorphism f

from a group G to a group H and bijective is called isomorphism. An

isomorphism f from a group G to itself is called automorphism. If

there is an isomorphism from G to H , we say that G and H are

isomorphic, and denoted by HG ≅ . We can summarize these as

follows:

a. Homomorphism f from G to G itself = endomorphism.

b. Homomorphism f from G to H and injective = monomorphism.

c. Homomorphism f from G to H and surjective = epimorphism.

d. Homomorphism f from G to H and bijective = isomorphism.



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e. Homomorphism f from G to H and surjective = epimorphism.

f. Isomorphism f from G to G itself = automorphism.

Note:

(i) if two groups are isomorphic, then they have the same cardinality.

(ii) if one group is cyclic, the other one must be cyclic, too. Suppose

that G is cyclic group generated by an element a , then H is

generated by ( )f a .

(iii) if one group is abelian, then the other one must be abelian as well.

Indeed, suppose H is abelian, then ( ) ( ) ( ) ( ) ( ) ( )f ab f a f b f b f a f ba= = =

and using the injectivity of f we get ab ba= . Two groups are

isomorphic, one may think that its elements are the same but they

have only different names.

Example 22: The additive group, ( , )+ of all integers is

isomorphic to the subgroup of even integers (2 , )+ ; that is, under the

isomorphism : 2 , 2f x x→ . It means they have the same

cardinality: 2= .

Example 23: Let *( , )⋅ be multiplicative group of all non-zero

complex numbers and *( , )⋅ be the multiplicative group of non-zero

real numbers. Let * *: ( , ) ( , ), ( )f z f z z⋅ → ⋅ = , then f is a homom-

orphism because for any *, ' : ( ') ' ' ( ) ( ')z z f zz zz z z f z f z∈ = = = .

Example 24: Let G be the group of real numbers under addition,

( , )+ , and H be the group of positive non-zero real numbers under

multiplication, *( , )+ ⋅ .

Let *: ( , ) ( , ), ( ) 2xf x f x++ → ⋅ = , then f is a homomorphism

because for any , : ( ) 2 2 2 ( ) ( )x y x yx y f x y f x f y+∈ + = = ⋅ = ⋅ .

In particular, if f is bijective, then *+≅ . On the other hand, if

*: , ( ) lnf x f x x+ → = , then f is bijective; that is, f is surjective

and injective. Since for any *, : ( ) lny yy x e f x e y+∈ ∃ = ∈ = = and for any



Introduction

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*1 2 1 2

1 2

1 2

, , ( ) ( )ln ln

x x f x f xx x

x x

+∈ =⇔ =⇒ =

Therefore, *+≅

Example 25: If 1,0 ≠> aa , then *: ( , ) ( , ), ( ) xf x f x a++ → ⋅ = is

an isomorphism, denoted by *+≅ .

DEFINITION 1.10: If HGf →: is a group homomorphism, then the

kernel of f is the set of all elements of G which maps into the identity

element He∈' . 1. ., ( ') { : ( ) '}i e Kernel of f f e x G f x e−= = ∈ = , denoted by

Kf =ker .

The image of f is the set ( ) { : ( ) },f G image of f y H y f x for some x G= = ∈ = ∈

denoted by fIm .

DEFINITION 1.11: The multiplicative group of all complex numbers

of absolute value 1, denoted by 1 { : 1 }z z= ∈ =S is called the group of

unit circle, i.e., the unit circle in the complex plane (see Definition 1.3

and Example 8 above).

Example 26: Using Example 23, we have 1 * 2 2(1 ) { : ( ) 1} { : 1}RK f z f z z x iy x y−= = ∈ = = = + + = which is the circle

with center 0 and radius 1.

THEOREM 1.12: If f is a homomorphism from a group G to a

group 'G with the identities e and 'e respectively, then

(i). ')( eef = and 11 )]([)( −− = xfxf .

(ii). )(Gf ; image of G , is a subgroup of 'G .

(iii). )'(1 efK −= is the subgroup of G .

Proof

(i). We have )(')()()()( efeefxfxefxf =⇔== by multiplying on

the left by 1)( −xf . In other words, if x has an inverse 1−x , then



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exxxx == −− 11 . So that 1 1 1 1' ( ) ' ( ) ( ) ( ) ( ) ( )e f e e f xx f x f x f x f x− − − −= ⇔ = = ⇔ = ;

multiply on the left by 1)( −xf . Therefore, 11 )]([)( −− = xfxf .

(ii). For any ')(, eefGe =∈ by (i), then )(Gfe∈ so that φ≠)(Gf .

For any )(',' Gfyx ∈ such that )('),(' yfyxfx == .

We have )()()()]()[('' 1111 −−−− === xyfyfxfyfxfyx .

Since Gxy ∈−1 , then )('' 1 Gfyx ∈− .

(iii). By (i) ')( eef = , then Kefe =∈ − )'(1 , so that φ≠fker .

For any fyx ker, ∈ satisfy )(')( yfexf == ,

then ''')]()[()()()( 111 eeeyfxfyfxfxyf ==== −−− .

Therefore, )'(11 efxy −− ∈ .



Main Theorem and Corollary

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2. OTHER DEFINITIONS, MAIN COROLLARY AND THEOREMS

THEOREM 2.1: Let f be a homomorphism from a group G to a

group 'G with the identities e and 'e respectively. Then

(i). f is injective if and only if }{ker ef = .

(ii). f is surjective if and only if 'Im Gf = .

Proof

(i). )(⇒ Assume that f is injective.

Then for any ')(,ker exffx =∈ . But ')( eef = by THEOREM 1.12 (i), so that

)()( efxf = . Since f is injective, then ex = . Therefore, }{ker ef = .

)(⇐ Assume that }{ker ef = . For any Gyx ∈, and if )()( yfxf = ,

then ')]()[()()()( 111 eyfxfyfxfxyf === −−− (by multiplying on the right by 1)( −yf ). It means that }{ker1 efxy =∈− so that yxexy =⇔=−1 (by

multiplying on the right by y ). Thus, f is injective.

(ii). )(⇒ Suppose that f is surjective.

Then for any ')(Im')(,,'' GGffxxfthatsuchGxGx ==⇒=∈∃∈ .

)(⇐ Suppose that '}:)({Im)( GGxxffGf =∈== . Then for any

')(:'', xxfGxGx =∈∃∈ . Thus, f is surjective.

DEFINITION 2.2: Let G be a group and H be a subgroup of G .

Then the left coset of H generated by x is }:{ HhxhxH ∈= . Similarly,

the right coset of H generated by x is of the form }:{ HhhxHx ∈= . In

particular, if the operation is addition, the left coset is defined by

}:{ HhhxHx ∈+=+ . Note that HHH = because if Hhh ∈', , then Hhh∈'

so that HHH ⊆ . Furthermore, if Hh∈ , then HHheh ∈= ' so that

HHH ⊆ .

THEOREM 2.3: Let G be a group and H be a subgroup of G , and

Gyx ∈, .

(i). .,.1 HxifonlyandifHxHparticularInHyxifonlyandifyHxH ∈=∈= −




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(ii). If yHxHthenyHxH =≠∩ ,φ .

(iii). .GxeveryforHxH ∈=

Proof

(i). First observe the relation on G and defined by

HyxifyxGyxallfor ∈≡∈ −1,, is an equivalence relation; that is,

. Reflexivity: Hexximpliesxx ∈=≡ −1 .

. Symmetry: Hyximpliesyx ∈≡ −1 , which implies Hyxxy ∈= −−− 111 )( .

This implies that xy ≡ .

. Transitivity: 1 1 1 1 1 1 1, ( )( ) ( )x y and y z imply x y H and y z H then x y y z x yy z x z− − − − − − −≡ ≡ ∈ ∈ = = .

This implies that zx ≡ . Therefore, ≡ is an equivalence relation on G .

Let }:{)(][ yxGyxclx ≡∈== be the equivalence classes, then

xHHhxhHhxhyGy

HhhyxGyHyxGy

yxGyxclx

=∈=∈=∈=∈=∈=

∈∈=

≡∈==

−

−

}:{},:{

},:{}:{

}:{)(][

1

1

These equivalence classes are the left coset of H in G . Furthermore,

the converses also hold. In particular, observe that 1 . , , .xH yH if and only if x y H Hence if y e then xH H if and only if x H−= ∈ = = ∈

(ii). Suppose , :xH yH then g G g xH yH g xH and g yHφ∩ ≠ ∃ ∈ = ∩ ⇒ ∈ ∈ .

Then there exist , ' 'h h H such that g xh and g yh∈ = = .

Thus, yHHhyhxHandhyhx === −− 11 '' , by (i).

(iii). Consider the map : , ( )f H xH h f h xh for all h H→ = ∈ .

Since xHHhxhHhhfHf =∈=∈= }:{}:)({)( , so it is surjective.

Furthermore, if for any '')'()(,', hhxhxhhfhfHhh =⇔=⇔=∈ , so it is

injective. Thus, f is bijective. So xHHxHequipotentH =⇒ .




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DEFINITION 2.4: G is a group and H is a subgroup of G . Then H

is called a normal subgroup of G if for any Gx∈ , HxxH = .

THEOREM 2.5: G is a group and H is a subgroup of G . Then H is

a normal subgroup of G if and only if for any Hh∈ and any Gx∈ ,

Hxhx ∈−1 . In particular, if G is an abelian group, then every subgroup

H is normal; since for any Hh∈ and any Gx∈ , then Hhhxxxhx ∈== −− 11 .

Proof

)(⇒ Suppose that H is a normal subgroup of G . Then for any

HxxHGx =∈ , . Then for any xHxhHh ∈∈ : . But HxxH = , then there

exists xhxhHh ':' =∈ . So that there exists '':' 11 hxxhxhxHh ==∈ −− . Thus,

Hxhx ∈−1 .

)(⇐ Suppose that for any Hh∈ and any HxhxGx ∈∈ −1, .

Then for any xHxh∈ there exists HxxhxhhxhxHh ∈=⇒=∈ − '':' 1 . Thus,

HxxH ⊂ . Similarly, assume that for any Hh∈ and any 1,x G x hx H−∈ ∈ .

Then for any Hxhx∈ , there exists xHxhhxhhxxHh ∈=⇒=∈ − '':' 1 .

Thus, xHHx⊂ . Therefore, HxxH = .

THEOREM 2.6: Let ': GGf → be a homomorphism, then fker is a

normal subgroup of G if and only if fxhx ker1 ∈− for any fhGx ker, ∈∈ .

Proof

)(⇒ Assume that fker is normal subgroup of G . If

Gxandfh ∈∈ ker , then 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ' ( ) ( ) ( ) ' kerf xhx f x f h f x f x e f x f x f x e xhx f− − − − −= = = = ⇒ ∈ .

)(⇐ Assume that H is a normal subgroup of G and that 'G is

the quotient group HG / . Define the map xHxxHGG =→ )(,/: ππ . π

is called the natural or canonical map .

Since )()())(()()( yxyHxHHxyxy πππ === , π is a homomorphism.

In other words, H is the identity element in HG / ,




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HHxGxHxHGxeHxGx =∈∈==∈==∈= }:{}:{})(:{ker ππ by THEOREM

2.3 (i).

Example 27: For any subgroup of abelian groups is normal.

Example 28: Given a group G with the identities e then }{e and

G itself are the normal subgroups of G . Since for any xeexGx }{}{, =∈

and 1−= xGxG .

Example 29: ( , )+ , the additive abelian group of all integers.

Then { : }n nk k= ∈ is normal subgroup of ( , )+ .

DEFINITION 2.7: Given G is a group and H is a normal subgroup

of G . Let }:{/ GxxHHG ∈= denote the family of all left cosets of H in

G with the operation that for any , / : ( )( ) ( )xH yH G H xH yH xy H∈ = .

Note that HG / is a group (left as exercise) called the quotient

group HbyG . The index of a subgroup H in G , denoted by ]:[ HG , is

the number of left (or right) cosets of H in G . Note also that when G is

finite, its order )/( HGo is the index )(/)(]:[ HoGoHG = that is why one

calls it the quotient group.

Example 30: Using Example 29, define a relation on by for

any , , (mod )x y x y n∈ ≡ if :x y nk k x y n− = ∈ ⇒ − ∈ . It is easy to see

that this is an equivalence relation on , and equivalence class of x is

[ ] { : (mod )}{ : , }{ : , }{ : }

x y y x ny y x nk ky y x nk kx nk k

= ∈ ≡= ∈ − = ∈= ∈ = + ∈= + ∈

This is called the residual class nmod .

Then [0] {0 , } { : }nk k nk k= + ∈ = ∈

[1] {1 , }nk k= + ∈

[2] {2 , }nk k= + ∈

[3] {3 , }nk k= + ∈

……………………

……………………




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Since for any ,l l nq r∈ = + such that , 0q r n∈ < < . Equivalently, for

any ,l l r nq∈ − = such that , 0 1q r n∈ < < − . Then for any

, (mod )l l r n∈ ≡ such that , 0 1q r n∈ < < − . So that for any

, [ ] [ ]l l r∈ ≡ such that 10 −<< nr . When we do the recursion on x , we

will get / {[0], [1], [2], , [ 1]}n n= − .

THEOREM 2.8: Given G is a group and H is a normal subgroup of

G . Then the map : / ,G G H x xHΦ → is epimorphism and H=Φker .

Proof

We have HGGxxHGxxG /}:{}:)({)( =∈=∈Φ=Φ . This is

surjective.

On the other hand, for any )()())(()()(:, yxyHxHHxyxyGyx ΦΦ===Φ∈ ;

Φ is a homomorphism. By DEFINITION 1.9, so that it is epimorphism.

Moreover,

ker { : ( ) } { : } { : } { : } .x G x eH x G xH eH x G xH H x G x H HΦ= ∈ Φ = = ∈ = = ∈ = = ∈ ∈ =

THEOREM 2.9: Given G is a group and H is a normal subgroup of

G and HGG /: →ψ is a canonical group epimorphism. For any group

'G and homomorphism ': GGf → such that fH ker⊂ , then

(i). there exists a unique group homomorphism '/: GHG →ϕ such

that fϕ ψ = .

(ii). ϕ is a group epimorphism if and only if f is a group

epimorphism.

(iii). ϕ is a group monomorphism if and only if fH ker= .

(iv). ϕ is a group isomorphism if and only if f is a group

epimorphism and fH ker= .

Proof

(i). Consider the correspondence : / ', ( ) ( )G H G xH xH f xϕ ϕ→ = .

ϕ is well-defined.




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1

1

1

1

1

kerker

( ) '( ) ( ) '

[ ( )] ( ) '( ) ( ) ( ).

If xH yHx y H fx y ff x y ef x f y ef x f y e

f x f y multiplying on the left by f x

−

−

−

−

−

=

⇔ ∈ ⊂

⇒ ∈

⇔ =

⇔ =

⇔ =⇔ =

Since HGyHxH /, ∈ ,

[( )( )] [( ) ] ( ) ( ) ( ) ( ) ( )xH yH xy H f xy f x f y xH yHϕ ϕ ϕ ϕ= = = = .

So ϕ is a homomorphism.

Thus, for any , ( )( ) ( ( )) ( ) ( )x G xH xH xH f xϕ ψ ϕ ψ ϕ∈ = = = . Furthermore,

suppose there is another '/:' GHG →ϕ such that ' fϕ ψ = .

Then for any / : ( ) ( ) ( ' )( ) '( ( )) '( )xH G H xH f x xH xH xHϕ ϕ ψ ϕ ψ ϕ∈ = = = = .

(ii). Since 'Im)(}:)({}/:)({)/(Im GfGfGxxfHGxHxHHG ===∈=∈== ϕϕϕ ,

so ϕ is surjective. Therefore, ϕ is a group epimorphism if and only if

f is a group epimorphism.

(iii). By THEOREM 2.1 (i). ϕ is injective (i.e., monomorphism) if

and only if }{ker H=ϕ .

We have HffxxHexfxHxH /ker}ker:{}')()(:{ker =∈==== ϕϕ .

Thus, ϕ is a group monomorphism if and only if Hf /ker consists only

of the identity element H ; i.e., HxHxHfx ∈⇔=⇒∈ ker .

Thus, ϕ is a group monomorphism if and only if Hf =ker .

(iv). By (ii) and (iii), we obtain that ϕ is a group isomorphism.

FIRST ISOMORPHISM THEOREM 2.10: Let G and H be groups.

If HGf →: is a group homomorphism, then ffG Imker/ ≅ .

Proof

Consider the correspondence ffG Imker/: →ψ

ker ( ker ) ( )x f x f f xψ =

is well-defined. Since if for any fGfyfx ker/ker,ker ∈ , then




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)()()()()]()[(

')()(')(

)(5.2kerkerker

1

1

1

1

xfyfxfyfxfxf

eyfxfdefinitionbyeyxf

iTheorembyfyxfyfx

=⇔=⇔

=⇔

=⇔

∈⇔

=

−

−

−

−

Since f is a homomorphism and ( ker ) ( ),x f f xψ =

[( ker )( ker )] ( ker ) ( ) ( ) ( ) ( ker ) ( ker )x f y f xy f f xy f x f y x f y fψ ψ ψ ψ= = = = .

So ψ is a homomorphism as well. If Im , ( )y f then y f x for some x G∈ = ∈

and so ( ) ( ker )y f x x fψ= = . Thus, ψ is surjective. In addition, if

( ker ) ( ker ), ( ) ( )x f y f then f x f yψ ψ= = .

Hence 1 1 1' ( ) ( ) ( ) ( ) ( )e f x f x f x f y f x y− − −= = = , so that fyfxfyx kerkerker1 =⇔∈−

by THEOREM 2.3 (i), so ψ is injective. So far, we have shown that ψ is an

isomorphism from ftofG Imker/ , i.e., ffG Imker/ ≅ .

Example 31: Using (Example 23) above, we obtain * * * */ ker Im { ( ) : } { : }f f f z z z z≅ = ∈ = ∈ =

COROLLARY 2.11: Let G and 'G be groups. If : 'f G G→ is a group

epimorphism, then 'ker/ GfG ≅ .

Proof

Using the FIRST ISOMORPHISM THEOREM 2.10, we get ffG Imker/ ≅ .

But f is a group epimorphism, then 'Im Gf = . Thus, 'ker/ GfG ≅ .

Next we want to see that the multiplicative groups giving some

examples of groups that are isomorphic to the subgroups of themselves.

We will consider the multiplicative groups of all positive rationals, + ;

non-zero rationals, * ; positive reals, + ; and non-zero reals, * .

THEOREM 2.12: The multiplicative groups , , , and+ ∗ + ∗

are all isomorphic to the proper subgroups of themselves.

Proof




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Consider * * 3: , ( )f x f x x→ = . f is homomorphism; that is,

for any * 3 3 3, , ( ) ( ) ( ) ( )x y f xy xy x y f x f y∈ = = = . f is injective ; that is, if

yxyxyfxf =⇒=⇔= 33)()( . Thus, f is isomorphic to the proper sub-

group of itself when restricted to + .

Next consider the function *: , xf x e→ . f is homomor-

phism; that is, for any , , ( ) ( ) ( )x y x yx y f x y e e e f x f y+∈ + = = = .

Furthermore, f is injective; that is, if yxeeyfxf yx =⇒=⇔= )()( .

Thus, f is isomorphic to the proper subgroup of itself

(see further details [6].).

THEOREM 2.13: There exists a group isomorphism : ( , ) ( , )f + → +

that extends the identity map of .

Proof

Extend {1} to a basis B of and a basis of as -vector

space. Then, we have respectively for any 1

, , ,n

i i i ii

x x e eλ λ=

∈ = ∈ ∈∑ B

and , ,

1, , ,

n

i í i ii

y y e eα α=

∈ = ∈ ∈∑ . Next we look for the cardinality of

1 1 2 21

#{ : } #{ : ... } #{ : ... }n

i i n ni

times

x x e x x e e e x xλ λ λ λ=

= = = = + + + = = + + + =∑B

B

and of

, , , ,1 1 2 2

1

#{ : } #{ : ... } #{ : ... }n

i í n ni

times

y y e y y e e e y yα α α α=

= = = = + + + = = + + + =∑

So we have = B and = . Since < and

max( , )=B B , we obtain = B . Similarly, = .

But 2= = × = × = because for any infinite cardinal

2,α α α= ; for example, if α is finite by taking {1, 2, 3},α = then




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we

see that 93 2 =≠= αα , and therefore = B .

Now choose a bijection :σ →B such that 1)1( =σ and extend it

to a vector space isomorphsm− (It has the same cardinality and is

bijective over ), :f → and then f is the required group

isomorphism such that ( ) ,f q q q= ∀ ∈ .

THEOREM 2.14: The multiplicative group of all non-zero complex

numbers, * is isomorphic to the multiplicative group of the unit

circle, 1S .

Proof

Consider the map *:ϕ →

izezz πϕ 2)( = is surjective.

We also see that ϕ is homomorphism; that is,

1 2 1 2 1 22 ( ) 2 2 2 21 2 1 2( ) ( ) ( )i z z i z i z i z i zz z e e e e z zπ π π π πϕ ϕ ϕ+ ++ = = = = . Moreover, ϕ is also

injective; that is, if 212122

21 22)()( 21 zzizizeezz zizi =⇒=⇔=⇔= ππϕϕ ππ .

By using the FIRST ISOMORPHISM THEOREM 2.10, we obtain */ kerϕ ≅ .

2 2

2

ker { : ( ) 1}{ : 1}{ : 0, 1}{ : 0, }

ix y

ix

But z zx iy e ex iy y ex iy y x

π π

π

ϕ ϕ−

= ∈ =

= + ∈ =

= + ∈ = == + ∈ = ∈=

Therefore, we obtain */ ≅ .

On the other hand, consider the map 1:ψ → S

rierr πψ 2)( =

is surjective. We see that ψ is homomorphism; that is,

)()()( 212222)(2

21212121 rreeeerr iriririrrri ψψψ πππππ ====+ ++ .

2 {(1,1), (1, 2), (1,3), (2,1), (2, 2), (2,3), (3,1), (3, 2), (3,3)},α α α= × =




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Applying the FIRST ISOMORPHISM THEOREM 2.10, we get 1/ kerψ ≅ S .

Moreover, we have 2ker { : 1}

{ : cos 2 sin 2 1}{ : cos 2 1}{ : cos 2 cos 0}{ : 2 2 , }{ : , }{ : }

irr er r i rr rr rr r k kr r k kr k

πψπ πππ

π π

= ∈ == ∈ + == ∈ == ∈ == ∈ = ∈= ∈ = ∈= ∈ ∈=

Thus, 1/ ≅ S .

By using the THEOREM 2.13; this therefore, induces that / /≅ .

In conclusion, we get * 1≅ S as required.



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3. BIBLIOGRAPHY

3.1 HISTORY OF GROUP DEVELOPMENT

The study of the development of a concept such as group is difficult. It must

be wrong to say that the set of non-zero integers under ordinary multiplication is a

group, since from the origin of a concept of group does not hold. Geometry has been

studied for a very long time ago so it is reasonable to ask what happened to geometry

at the beginning of the 19th Century that was to contribute to the rise of the group

concept.

In 1761, Euler studied modular arithmetic. In particular, he examined the

remainders of powers of a number modulo n. although Euler’s work is, of course, not

stated in group theoretic terms he does provide an example of the decomposition of an

abelian group into cosets of a subgroup. He also proves that, a special case of, the

order of a subgroup is a divisor of the order of the group.

In 1799, Ruffini, the first person who claimed that the equations of degree 5

could not be solved algebraically and he published a work whose purpose was to

demonstrate the insolubility of the general quintic equation; Ruffini’s work is based

on proof of Lagrange’s but Ruffini introduces groups of permutations. He calls

permutazione and explicitly uses the closure property (the associative law always

holds for permutations). Ruffini divides his permutazione into two kinds, namely

permutazion semplice which are cyclic group in modern notation, and permutazione

composta which are non-cyclic group. Ruffini’s proof of insolubility of the quintic

equation has some gaps and disappointed with the lacks of reaction to his paper. He

published futher proofs (in 1802) showing that the group of permutations associated

with an irreducible equation is transitive taking his understanding well beyond that of

Lagrange.

In 1801, Gauss was to take Euler’s work much further and give a considerable

amount of work on modular arithmetic which amounts to a fair amount of the theory

of abelian groups. He examines the orders of elements and proves that there is a

subgroup for every number dividing the order of a cyclic group. Gauss also examined

other abelian groups and has a finite abelian group and later (in 1869) Schering, who

edited Gauss’s works, found a basis for this abelian.



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In 1815, Cauchy and his paper on the subject played an important role in

developing the theory of permutation but at this stage Cauchy is motivated by

permutations of roots of equation.

In 1824, Abel gave the first accepted proof of the insolubility of the quintic

equation, and he used the existing ideas on permutations of roofs but entitle new in

the development of the group theory.

In 1827, Möbius began to classify geometries using the pact that a particular

geometry studies the properties invariant under a particular group although he was

totally unaware of the group concept.

In 1831, Galois was the first to really understand that the algebraic solution of

an equation was related to a structure of a group le groupe of permutations related to

the equation. By 1832, Galois had discovered that special subgroups (now called

normal subgroups) are fundamental. He calls the decomposition of a group into cosets

of a subgroup a proper decomposition if the right and left coset decompositions

coincide. Galois then shows that the non-abelian simple of smallest order has order 60.

In 1832, Steiner studied notion of synthetic geometry that were to eventually

become part of the study of transformation groups. On the other hand, in 1844,

Cauchy published a main work which sets up the theory of permutations as a subject.

He introduces the notation of powers, positive and negative, of permutations (with the

power zero giving the identity permutation), defines the order of a permutaion,

introduces cycle notation and used the term syste`me des substiutions conjugees.

Cauchy calls two permutations similar if they have the same cycle structure and

proves that this is the same as the permutations being conjugate. Galois’s work was

not known until Liouville published Galois’s papers in 1846. Liouville saw clearly

the connection between Cauchy’s theory of permutation and Galois’s work. However,

Liouville failed to grasp that the importance of Galois’s work lay in the group concept.

It was due to the English mathematician Cayley, perhaps the most remarkable

development groups had become even before Betti’s work. As early as 1849, Cayley

published a paper linking his ideas on permutations with Cauchy’s.

In 1851, Betti began publishing work relating permutation theory and the

theory of equations. In fact Betti was the first to prove that Galois’s group associated

with an equation was in fact a group of permutations in modern sense. Serret



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published an important work discussing Galois’s work, still without seeing the

significance of the group concept.

In 1854, Cayley wrote papers which are remarkable for the insight they have

of abstract groups. At that time the only famous groups were groups of permutations

and even this was a new area, because Cayley defines an abstract group and gives a

stable to display the group multiplication. He gives the “ Cayley Table” of some

special permutation groups, but much more significantly for the introduction of the

abstract group concept, he realized

that matrices and quaternions were groups. Cayley’s papers of 1854 were so far ahead

of their time that they had little impact. However, when Cayley returned to the topic

in 1878 with four papers on groups, one of them called “the theory of groups”, the

time was right to the abstract group concept to move forwards the centre of

mathematical investigation. Cayley proved that every finite group can be represented

as a group of permutations. Cayley’s work promoted Hölder in 1893, to investigate

the groups of order 423 ,,, pandpqrpqp . Frobenius and Netto (a student of

Kronecker) carried the theory of groups forwards. As far as the abstract group concept

is concerned, the next major contributor was von Dyck , who had obtained his

doctorate under Klein’s supervision then became Klein’s assistant. von Dyck , with

fundamental papers in 1882 and 1883, constructed free groups and the definition of

abstract groups in terms of generator and relations.

Jordan, however, in the papers of 1865, 1869, and 1870 shows that he

realizes the significance of groups of permutations. He defines isomorphism of

permutation groups and proves the Jordan-Holder theorem for permutation groups.

Holder was to prove it in the context of abstract groups in 1889.

Klein proposed the Erlangen Program in 1872 which was the group theoretic

classification of geometry. Groups were certainly becoming centre stage in

mathematics.

Group theory really came up and came of age with the by Burnside Theory of

groups of finite order published in 1897. The two volume algebra books by Heinrich

Weber (a student of Dedekind) Lehrbuch der Algebfra published in 1895 and 1896

became a standard text. These books influenced the next generation of



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mathematicians to bring group theory into the most major theory of the 20th Century

mathematics.

(Article by J. J. O’Connor and E. F. Robertson)



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3.2 REFERENCES

[1]. J. R. Clay, “The Punctured Plane is Isomorphic to the Unit

Circle", Journal of Number Theory 1, 500-501, 1969

[2]. H. Azad and A. Laradji, “On a Theorem of Clay”, College

Mathematics Journal, Vol. 31, No. 5, 405-406, 2000

[3]. J. A. Gallian, Contemporary Abstract Algebra, 3rd edition, D.C.

Health and Company, Toronto, 1994

[4]. J. J. Rotman, Advanced Modern Algebra, Prientice Hall, 2nd

printing 2003

[5]. J. Fang, Abstract Algebra, Schaum Publishing Company, 1963

[6]. Shaun Fallat, Chi-Kwong Li, David Lutzer and David Stanford,

“On the groups that are isomorphic to a proper subgroup”,

Mathematics Magazine, Vol.72, No.5, 388-391, December, 1999

[7]. Seymour Lipschutz, Theory and Problems of Linear Algebra,

McGraw- Hill, Inc, 1968

[8]. Robert B. Ash, Abstract Algebra: The Basic Graduate Year,

© October 2000

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