the relative commutativity degree and sub...

THE RELATIVE COMMUTATIVITY DEGREE AND SUB-MULTIPLICATIVE DEGREE

FOR NONCYCLIC SUBGROUPS OF SOME NONABELIAN METABELIAN GROUPS

FADHILAH BINTI ABU BAKAR

UNIVERSITI TEKNOLOGI MALAYSIA

THE RELATIVE COMMUTATIVITY DEGREE AND SUB-MULTIPLICATIVE DEGREE

FOR NONCYCLIC SUBGROUPS OF SOME NONABELIAN METABELIAN GROUPS

FADHILAH ABU BAKAR

A dissertation submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Science

Faculty of Science

Universiti Teknologi Malaysia

APRIL 2017

iii

To my beloved Ummi and Abah

iv

A C K N O W LED G EM EN TS

In the name of Allah, the Most Gracious and Merciful. He has given me

strength and courage to accomplish this dissertation.

I would like to take this opportunity to express my utmost gratitude to

everyone involved in completing this dissertation. High appreciation to my dissertation

supervisor, Dr. Nor Muhainiah Mohd Ali for her effort and patience in guiding me to

conduct this research. Thanks a lot for all the encouragement, critisms and suggestions.

I am also indebted to M inistry of Higher Education (M OHE) Malaysia for

sponsoring my master study through M yBrain’ 15 scholarship. My special thanks to my

family especially my parents, Zubidah binti Abu Bakar and Abu Bakar bin Mohammad

for their full encouragement, support and inspiration while completing this dissertation.

Last but not least, I would like to thank my close friends, Fasihah Zulkiflee and Amira

Fadina Ahmad Fadzil for their assistance and encouragement throughout the process

of completing this dissertation.

v

A BSTRACT

A metabelian group is a group G that has at least an abelian normal subgroup N

such that the quotient group G/n is also abelian. The concept of commutativity degree

plays an important role in determining the abelianness of the group. This concept has

been extended to the relative commutativity degree of a subgroup H of a group G

which is defined as the probability that an element of H commutes with an element

of G. This notion is further extended to the notion of the multiplicative degree of

a group G which is defined as the probability that the product of a pair of elements

chosen randomly from a group G is in the given subgroup of H . By using those two

definitions with an assistance from Groups, Algorithms and Programming and Maple

software, the relative commutativity degree and sub-multiplicative degree for noncyclic

subgroups of nonabelian metabelian groups of order less than 24 and dihedral groups

of order at most 24 are determined in this dissertation.

vi

A BSTRA K

Suatu kumpulan metabelan adalah kumpulan G yang mengandungi sekurang-

kurangnya satu subkumpulan normal abelan N dengan syarat kumpulan pembahagi

G/ n juga adalah abelan. Konsep darjah kekalisan tukar tertib memainkan peranan

penting dalam menentukan keabelan bagi suatu kumpulan. Konsep ini telah dilanjutkan

kepada darjah kekalisan tukar tertib relatif bagi suatu subkumpulan H bagi suatu

kumpulan G yang ditakrifkan sebagai kebarangkalian bahawa suatu unsur H berkalis

tukar tertib dengan suatu unsur dari G. Konsep ini diperlanjutkan kepada konsep darjah

pendaraban yang ditakrifkan sebagai kebarangkalian bahawa hasil darab sepasang

unsur yang dipilih secara rawak dari kumpulan G, berada dalam subkumpulan H .

Dengan menggunakan kedua-dua takrifan tersebut beserta bantuan daripada perisian

Groups, Algorithms and Programming dan Maple, darjah kekalisan tukar tertib relatif

dan darjah sub-pendaraban bagi subkumpulan tidak kitaran bagi kumpulan metabelan

yang tidak abelan bagi peringkat kurang daripada 24 dan kumpulan dwihedron bagi

peringkat sekurang-kurangnya 24 telah ditentukan di dalam disertasi ini.

vii

C H A PT E R T IT L E PAGE

D ECLA RA TIO N ii

D ED ICATIO N iii

A C K N O W LED G EM EN TS iv

A BSTRACT v

A BSTRA K vi

TABLE O F CO N TEN TS vii

L IST O F TABLES xi

L IST O F SYM BOLS xii

1 IN TR O D U C TIO N 1

1.1 Introduction 1

1.2 Research Background 2

1.3 Problem Statement 4

1.4 Research Objectives 4

1.5 Scope of the Study 5

1.6 Significance of Findings 5

1.7 Research Methodology 6

1.8 Dissertation Organization 7

2 LITER A TU R E R EV IEW 9

2.1 Introduction 9

2.2 Metabelian Groups 9

2.3 The Commutativity Degree of a Group 12

TABLE OF CONTENTS

viii

2.4 The Relative Commutativity Degree of a Subgroup

of a Group 15

2.5 Sub-Multiplicative Degree of a Group 16

2.6 Groups, Algorithms and Programming (GAP) Software 17

2.7 Conclusion 18

3 N O N C Y C LIC SUBGROUPS O F SO M E

N ON A BELIAN M ETA BELIA N G ROU PS

3.1 Introduction

3.2 Steps to Find Noncyclic Subgroups by Using GAP

Software

19

19

2 0

3.2.1 Library of GAP Software for Dihedral,

Alternating and Symmetric Group 2 0

3.2.2 Using Group Presentation and Group ID 2 1

Noncyclic Subgroups of Nonabelian Metabelian

Groups of Order Less Than 24 23

3.3.1 Noncyclic Subgroup of A 4 23

3.3.2 Noncyclic Subgroups of Quasihedral-16 23

3.3.3 Noncyclic Subgroups of Q 8 24

3.3.4 Noncyclic Subgroups of D 4 x Z 2 24

3.3.5 Noncyclic Subgroups of Q x Z 2 25

3.3.6 Noncyclic Subgroups of Modular-16 25

3.3.7 Noncyclic Subgroups of B 26

3.3.8 Noncyclic Subgroups of K 26

3.3.9 Noncyclic Subgroups of G 4 ,4 27

3.3.10 Noncyclic Subgroups of S 3 x Z 3 27

3.3.11 Noncyclic Subgroups of (Z 3 x Z 3 ) x Z 2 28

3.3.12 Noncyclic Subgroup of Z 5 x Z 4 28

Noncyclic Subgroups of Dihedral Groups of Order

at M ost 24 29

3.4.1 Noncyclic Subgroups of D 4 29


ix



3.4.5 Noncyclic Subgroups of D i 0 31

3.4.6 Noncyclic Subgroups of D 1 2 31

3.5 Conclusion 33

4 RELATIV E CO M M UTATIV ITY D E G R E E O F

SO M E NON ABELIAN M ETA BELIA N G ROU PS 34

4.1 Introduction 34

4.2 The Relative Commutativity Degree for Noncyclic

Subgroups of Nonabelian Metabelian Groups of

Order Less Than 24 35

4.3 The Relative Commutativity Degree for Noncyclic

Subgroups of Dihedral Groups of Order at Most 24 59

4.4 Conclusion 69

5 SUB -M ULTIPLICA TIVE D E G R E E O F SO M E

N ON A BELIAN M ETA BELIA N G RO U PS 70

5.1 Introduction 70

5.2 The Sub-Multiplicative Degree for Noncyclic

Subgroups of Nonabelian Metabelian Groups of

Order Less Than 24 71

5.3 The Sub-Multiplicative Degree for Noncyclic

Subgroups of Dihedral Groups of Order at Most 24 97

5.4 Conclusion 110

6 C O N C LU SIO N 111

6 . 1 Introduction 1 1 1

6.2 Summary 111

6.3 Suggestions for Further Research 114

R EFER E N C E S 114

A PPEN D IC ES 117

x

A Publication an d P resen tation in Conferences 118

A.1 Non-Indexed Proceedings 118

A.2 Papers Presented in Conference 118

A.3 Abstract Accepted in Conference 119

B Cayley Table of N onabelian M etabelian G roups of

O rd e r Less T han 24 120

C Cayley Table of D ihedral G roups of O rd e r a t M ost 24 131

xi

TABLE NO.

2 . 1

2 . 2

2.3

2.4

3.1

3.2

4.1

4.2

4.3

6 . 1

6 . 2

6.3

LIST OF TABLES

T IT L E

The nonabelian metabelian groups of order less than 24

Dihedral groups of order at most 24

The nonabelian metabelian groups of order less than 24

that have noncyclic subgroups

Dihedral groups of order at most 24 that have noncyclic

subgroups

Number of noncyclic subgroups of nonabelian

metabelian groups of order less than 24

Number of noncyclic subgroups of dihedral groups of

order at most 24

The Cayley table of A 4

The 0-1 Table of A 4

The 0-1 Table for multiplication between elements in

{e, a, b, ab] and elements of A 4

Relative commutativity degree for noncyclic subgroups

of nonabelian metabelian groups of order less than 24

Relative commutativity degree for dihedral groups of

order at most 24

The sub-multiplicative degree for noncyclic subgroups

of some nonabelian metabelian groups

PAGE

1 0

1 1

1 1

1 2

29

32

36

36

37

1 1 2

113

113

xii

LIST OF SYMBOLS

Z (G ) - Center of a group G

P (G) - Commutativity degree of a group G

[hn , g] - Commutator of two elements, hn and g

X - Direct product

G - Element of

= - Equal to

V - For all

> - Greater than or equal to

(x) - Group generated by the element x

hn - n - 1 hn, conjugate element

e, 1 - Identity element in a group

= - Isomorphic

Pmul (G) - Multiplicative degree of a group G

N - Normal subgroup

^ - Not a subset of

G - Not an element of

= - Not equal to

Pn (G) - n-th commutativity degree of G

| G| - Order of the group G

| H | - Order of the subgroup H

G/n - Quotient group

P (H, G) - Relative commutativity degree of G

xiii

Pn (H, G) - Relative n-th commutativity degree of G

Ps-ubm (H, G) - Sub-Multiplicative degree of a group G

C - Subset

x G G \H - x is an element of G but not in H

CHAPTER 1

IN TR O D U C TIO N

1.1 In troduction

Probability theory is the mathematical study of uncertainty. Every day we face

some situations where the result is uncertain, and perhaps without realizing it, we guess

about the likelihood of one outcome or another. These days, probability theory plays

an increasingly greater role in many fields. Algebra and analysis is one of the fields

that take probabilistic theory importantly.

A metabelian group is a group G that has at least a normal subgroup N such

that N and G/ n are both abelian. A metabelian group can be considered as a group that

is closed to being abelian, in the sense that every abelian group is metabelian, but not

all metabelian groups are abelian. By the definition of metabelian group, any dihedral

group, D n is metabelian since it has a cyclic normal subgroup, N and the quotient

group of D n /N is a cyclic group of order two, hence abelian.

Let a and b be elements in a group G. Then the commutativity degree of a group

G is the probability of a random pair of elements in G commute. It is also equivalent as

the number of pairs (a,b) such that ab = ba, divided by the total number of all possible

pairs of (a,b).

2

By extending the notion ab = ba to anb = ban, the n th commutativity degree

of a group G can be found. By considering the total number of pair (a,b) for which an

and b commute i.e. anb = ban, divided by the total number of possible pairs of (a,b).

Thus the n th commutativity degree of a group is determined.

Next, the concept of relative commutativity degree of a subgroup H of a group

G is defined as the extension of the concept of commutavity degree. The relative

commutativity degree is defined as the probability for an element of H to commute

with an element of G.

The notion of commutativity degree has lead to the concept of multiplicative

degree of a group G which is defined as the probability that the product of a pair of

elements chosen randomly from a group G is in the given subgroup H of G. There

are four cases that can be considered in the concept of multiplicative degree which are

x, y G H , x G G \H but y G H , x G H but y G G \H and lastly x, y G G \H . For

the case x ,y G G \H , we defined it as the sub-multiplicative degree where it is the

probability that the product of a pair of elements chosen randomly from a group G but

not in H is in the given subgroup H of G. In this research, the relative commutativity

degree and sub-multiplicative degree of nonabelian metabelian groups of order less

than 24 and dihedral group of order at most 24 are determined where H is their

noncyclic subgroup.

1.2 R esearch B ackground

Metabelian group can be considered as a group close to being abelian, in the

sense that every abelian group is metabelian, but not all metabelian groups are abelian.

Metabelian can be described as Abelian-by-Abelian. Abdul Rahman et al. [1] has

found all the metabelian groups of order at most 24 in 2012.

3

First, let a and b be elements in a group G. Then the commutativity degree of a

group G is the total number of a random pairs of elements in G that commute divided

by the total number of possible pairs of elements in G. M iller [2] was the first person

who introduced the concept of commutativity degree of a finite group in his paper in

1944.

In 2006, Mohd Ali and Sarmin [3] extended the notion of commutativity degree

of finite group G by defining the n th commutativity degree of G denoted by Pn (G). The

n th commutativity degree is the probability that the n th power of a random element

commutes with another random element from the same group.

In 2007, Erfanian et al. [4] introduced the concept of relative commutativity

degree of a subgroup H of a group G, denoted as P (H ,G ). The relative commutativity

degree of a subgroup H , which is the probability for an element of H to commute with

an element of G, is the extension of the concept of commutativity degree.

In early 2016, Abd Rhani et al. [5] introduced the multiplicative degree of a

group G denoted as Pmul (G). This is the further extension of the concept of relative

commutativity degree. It is defined as the probability that the product of a pair of

elements chosen randomly from a group G is in the given subgroup H of G. There

are four cases that can be considered which are x, y G H , x G G \H but y G H ,

x G H but y G G \H and lastly x, y G G \H . For the case x, y G H , then we haveIH |2Pmui (G) = jgja-. For the case x , y G G \H , it is defined as the sub-multiplicative degree

which is the probability that the product of a pair of elements chosen randomly from a

group G but not in H is in the given subgroup H of G.

Previously, the commutativity degree of nonabelian metabelian groups of order

at most 24 has been determined by Che Mohd [6 ] in 2011. Two years later, Abd

Halim and Mohd Ali [7] determined Pn (G) for some nonabelian metabelian groups.

The relative commutativity degree of a cyclic subgroup of metabelian group of order

4

at most 24 has been determined by Hassan in [8 ] and recently, the sub-multiplicative

degree for cyclic subgroups of nonabelian metabelian groups of order less than 24 is

determined by Mustafa et al. in [9] and Jaafar et al. in [10].

In this research, the relative commutativity degree and multiplicative degree

for noncyclic subgroups of nonabelian metabelian groups of order less than 24 and

dihedral groups of order at most 24 are determined.

1.3 P roblem S tatem ent

Previous studies have been done in determining the relative commutativity

degree and sub-multiplicative degree for various groups. However, the relative

commutativity degree and sub-multiplicative degree for noncyclic subgroups of

nonabelian metabelian groups of order less than 24 and dihedral groups of order at most

24 have not been done yet. Previous researchers only focus on cyclic subgroups of the

groups. Thus problem that arises here is what is the relative commutativity degree and

sub-multiplicative degree for noncyclic subgroups of nonabelian metabelian groups of

order less than 24 and dihedral groups of order at most 24.

1.4 R esearch O bjectives

The objectives of this research are:

1. To determine the noncyclic subgroups of nonabelian metabelian groups of

order less than 24 and dihedral groups of order at most 24.

2. To find the relative commutativity degree for noncyclic subgroups of

nonabelian metabelian groups of order less than 24 and dihedral groups of

order at most 24.

5

3. To determine the sub-multiplicative degree for noncyclic subgroups of

nonabelian metabelian groups of order less than 24 and dihedral groups of

order at most 24.

1.5 Scope of the S tudy

This research consists of three parts. The first part focuses only on noncyclic

subgroups of nonabelian metabelian groups of order less than 24 and dihedral groups of

order at most 24. The second part of this research focuses on the relative commutativity

degree for noncyclic subgroups of nonabelian metabelian groups of order less than 24

and dihedral groups of order at most 24. The last part covers on the sub-multiplicative

degree for noncyclic subgroups of nonabelian metabelian groups of order less than 24

and dihedral groups of order at most 24.

1.6 Significance of F indings

The results of this research can be used for further research in related areas

such as to find other generalized commutativity degree involving noncyclic subgroups

of nonabelian metabelian groups of order less than 24 and dihedral groups of order at

most 24. In addition, this research will lead to the production of technical reports and

papers published in an open literature.

1.7 R esearch M ethodology

6

This research begins by studying the various concepts and properties of the

nonabelian metabelian groups of order less than 24 and dihedral groups of order

at most 24. To find the Cayley table of respective groups, Maple Software [29]is

use.Next, the noncyclic subgroups of those groups are determined by the assistance

of Groups, Algorithms and Programming (GAP) Software [11]. Then, the concept

of commutativity degree is studied to have more understanding on the relative

commutativity degree and sub-multiplicative degree. Next, the relative commutativity

degree are computed for noncyclic subgroups of nonabelian metabelian groups of order

less than 24 and dihedral groups of order at most 24. Lastly, the sub-multiplicative

degree are computed for the same groups. Figure 1.1 shows the methodology of this

dissertation

F igure 1.1 Research Methodology

7

1.8 D issertation O rgan ization

This thesis is divided into six chapters which includes the introduction,

literature review, the noncyclic subgroups of some nonabelian metabelian groups,

the relative commutativity degree of some nonabelian metabelian groups, the sub-

multiplicative degree of some nonabelian metabelian groups and lastly the conclusion.

The first chapter serves as an introduction to the whole dissertation including

the research background, problem statements, research objectives, scope of the study,

significance of finding, research methodology and dissertation organization.

In Chapter 2, the literature review of this research is presented. In this chapter, a

brief history about commutativity degree is stated in the background of the study, where

some earlier and recent works related to commutativity degree and its generalizations

namely the relative commutativity degree, n th commutativity degree and the sub-

multiplicative degree are provided. Some definitions and basic concepts that are used

throughout this research are also included.

All noncyclic subgroups of all nonabelian metabelian groups of order less than

24 and dihedral groups of order at most 24 are presented in Chapter 3. Groups,

Algorithms and Programming (GAP) Software [11] is used to assist in the finding of

those noncyclic subgroups.

Chapter 4 and 5 discuss on the findings of this dissertation. Chapter 4

includes the results on the relative commutativity degree for noncyclic subgroups of

all nonabelian metabelian groups of order less than 24 and dihedral groups of order at

most 24. Chapter 5 discusses on the findings and the results on the sub-multiplicative

degree for noncyclic subgroups of all nonabelian metabelian groups of order less than

24 and dihedral groups of order at most 24.

8

Lastly, Chapter 6 is the conclusion chapter where the summary and some useful

suggestions for future research are included. Published papers in conferences are listed

in Appendix A. The Cayley tables and 0-1 Table for all nonabelian metabelian groups

of order less than 24 with the exception of dihedral groups are included in Appendix

B, and the Cayley tables and 0-1 Table for all dihedral groups of order at most 24 are

included in Appendix C.

R EFER E N C E S

1. Abdul Rahman, S. F. and Sarmin, N. H., Metabelian Groups of Order at Most 24.

Discovering Mathematics, 2012. 34(1):77-93.

2. Miller, G. A. Relative Number of Non-invariant Operators in a Group. Proc. Nat.

Acad. Sci. USA, 1944. 30(2):25-28.

3. Mohd Ali, N. M. and Sarmin, N. H., On Some Problem in Group Theory of

Probabilistic Nature. Technical Report o f Department o f Mathematics, Universiti

Teknologi Malaysia, LT/M Bil.8/2006.

4. Erfanian, A., Rezaei, R., and Lescot, P. On the Relative Commutativity Degree of

a Subgroup of a Finite Group Comm. Algebra, 2007. 35(12): 4183-4197.

5. Abd Rhani,N., Mohd Ali,N. M., Sarmin, N. H., Erfanian, A. and Abdul Hamid,M.,

Multiplicative Degree of Some Dihedral Groups, AIP Conference Proceedings,

2016. 1750: 050003.

6. Che Mohd, M. The Commutativity Degree o f All Nonabelian Metabelian Groups

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7. Abd Halim, Z. and Mohd Ali, N. M. The Probability of Degree for some

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837.

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Nonabelian Metabelian Groups o f Order at most 24. Universiti Teknologi

Malaysia: Masters Thesis. 2014.

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Metabelian Groups of Order Less Than 24 Except 16. Prosiding Projek Sarjana

Muda Sains. Universiti Teknologi Malaysia. 2015/2016(2): 247-257.

116

10. Jaafar, N. A. and Mohd Ali, N. M. The Multiplicative Degree Nonabelian

Metabelian Groups of Order 166. Prosiding Projek Sarjana Muda Sains. Universiti

Teknologi Malaysia. 2015/2016(1): 170-174.

11. The Groups, Algorithms, and Programming, (GAP), Version 4.8.6:

http://www.gap-system.org. 2016.

12. Wisnesky, R.J. Solvable Groups. Math 120. 2005. wisnesky.net/wim3.pdf

13. Gustafson, W. H. W hat is the probability that two elements commute? American

Mathematical Monthly , 1973.08(9)(9): 1031-1034.

14. Machale, D. How commutative can a non-commutative group be? The

Mathematical Gazette, 1974. 58(405): 199-202.

15. Erdos, P. and Turan, P. On some problems of a statistical group theory. IV Acta

Math. Acad Sci. Hungaricae, 1968. 19: 413435.

16. Sherman, G., The Lower Bound fo r the Number o f Conjugacy Classes in Finite

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Arch.Math, 2009. 93: 345-356.

18. Moradipour, K., Sarmin, N. H. and Erfanian, A., The Precise Value o f

Commutativity Degree in Some Finite Groups. Malaysian Journal of Fundamental

and Applied Sciences, 2012. 8(2): 67-72.

19. Mohammed Hussein Saleh, H. The Commutativity Degree o f Some Nonabelian

Two-Groups with a Cyclic Subgroup o f Index Four. Universiti Teknologi Malaysia:

Master Degree Project Report. 2014.

20. Abdul Hamid, M. The Probability That Two Elements Commute in Dihedral

Groups. Universiti Teknologi Malaysia: Undergraduate Project Report. 2011.

21. Abu Sama, N The Commutativity Degree o f Alternating Groups up to Degree 200.

Universiti Teknologi Malaysia: Undergraduate Project Report. 2010.

22. Yahya, Z. The nth Commutativity Degree o f Dihedral Groups up to Degree 12.

Universiti Teknologi Malaysia: Undergraduate Project Report. 2011.

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117

23. Abu Bakar, F. and Mohd Ali, N. M., The nth Commutativity Degree o f Nonabelian

Groups o f Order 16 With a Cyclic Subgroup o f Index Four. Proceedings of 4th

International Science Postgraduate Conference, 2016.: 156-160.

24. Abd Manaf, F. N., Sarmin, N. H., Mohd Ali, N. M. and Erfanian, A., The nth

Commutativity Degree o f Some Finite Groups o f Nilpotency Class Two. Extended

abstract of the 22nd Iranian Algebra Seminar, 2012. : 17-20.

25. Barzgar, R., Erfanian, A. and Farrokhi, M. D. G., Finite Groups with Three

Relative Commutativity Degrees. Bulletin of the Iranian Mathematical Society,

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26. Erfanian, A., Tolue, B., and Sarmin, N. H., Some Considerations on the nth

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the relative commutativity degree and sub...

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