GROUPS OF ORDER AT MOST 24

THE nth COMMUTATIVITY DEGREE OF N ONABELIAN MET ABELIAN GROUPS OF ORDER AT MOST 24 ZULEZZAH BINTI ABD HALIM UNIVERSITI TEKNOLOGI MALAYSIA

THE nth COMMUTATIVITY DEGREE OF NONABELIAN MET ABELIAN GROUPS OF ORDER AT MOST 24 ZULEZZAH BINTI ABD HALIM A dissertation submitted in partial fulfillment of the requirements for the award of the degree of Master of Science (Mathematics) Faculty of Science Universiti Teknologi Malaysia JANUARY 2013

To my beloved mother, father and family lll

iv ACKNOWLEDGEMENTS First of all, thanks to Allah, who has given me the strength to complete this dissertation. In preparing this thesis, I was in contact with many people. They have contributed towards my understanding and thoughts. In particular, I wish to express my sincere appreciation to my dissertation supervisor, Dr. Nor Muhainiah Mohd Ali for her encouragement, guidance and invaluable suggestions. She has helped me in many ways besides making sure that I had all necessary information as a reference to complete this dissertation. Moreover, I would like to express my thanks to my parents, Abd Halim Othman and Zakiah Wan Abdullah for their long lasting love. Lastly, I would like to thank my siblings, my friends and Mohd Efiez Mustafa. They really supported me and played an important role in the completion of my thesis. I would like to thank them for their encouragement, love and emotional support.

v ABSTRACT A group G is metabelian if and only if there exists an abelian normal subgroup A such that the factor group, G/j^ is abelian. Meanwhile, for any group G, the commutativity degree of a group is the probability that two randomly selected elements of the group commute and denoted as P(G). Furthermore, the nth commutativity degree of a group G is defined as the probability that the nth power of a random element commutes with another random element from the same group, Pn(G). In this research, P(G) and Pn(G) for nonabelian metabelian groups of order up to 24 are computed and presented. The nth commutativity degree of a group are found by using the formula of Pn(G).

vi ABSTRAK Suatu kumpulan G adalah metabelan jika dan hanya jika wujud satu subkumpulan normal yang abelian, A dengan syarat kumpulan faktornya, G/^ adalah abelian. Sementara itu, bagi suatu kumpulan G, darjah kekalisan tukar tertib bagi kumpulan itu ialah kebarangkalian dua unsur dipilih secara rawak dari kumpulan itu adalah berkalis tukar tertib dan ditandakan sebagai P(G). Tambahan lagi, darjah kekalisan tukar tertib ke-n ditakrifkan sebagai kebarangkalian kuasa ke-n bagi suatu unsur rawak berkalis tukar tertib dengan suatu unsur rawak yang lain dari kumpulan yang sama, P n(g). Dalam kajian ini, P(G) dan P n(g) bagi kumpulan metabelan tak abelan dihitung dan diperkenalkan. Darjah kekalisan tukar tertib ke-n dicari dengan menggunakan formula P n(g).

vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENTS ABSTRACT ABTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF SYMBOLS ii iii iv v vi vii xi xii 1 INTRODUCTION 1 1.1 Introduction 1 1.2 Research Background 2 1.3 Problem Statement 3 1.4 Research Objective 3 1.5 Scope of Research 3 1.6 Significance of Study 4 1.7 Thesis Organization 4 1.8 Conclusion 5 2 LITERATURE REVIEW 6 2.1 Introduction 6 2.2 Metabelian Groups 6 2.2.1 Some Basic Concepts and Properties 7

viii in Metabelian Groups 2.3 Commutativity Degree of Groups 9 2.3.1 Definitions of Terms 12 2.4 The nth Commutativity Degree of a Group 13 2.5 Conclusion 15 3 THE nth COMMUTATIVITY DEGREE OF 16 NONABELIAN METABELIAN GROUPS OF ORDER LESS THAN 24 3.1 Introduction 16 3.2 Finding the nth Commutativity Degree 16 Using 0-1 Table 3.3 Some Results on Pn(G) 17 3.3.1 Pn(D3) 17 3.3.2 Pn(D4) and P n Q ) 18 3.3.3 Pn(D 5) 19 3.3.4 Pn(Z 3 x Z 4) 19 3.3.5 Pn(A4) 2 5 3.3.6 Pn(D 6) 2 7 3.3.7 Pn(D 7) 2 7 3.3.8 Pn(D 8) 2 8 3.3.9 P n(quasihedral-16) 3 1 3.3.10 Pn(Qg) 3 3 3.3.11 Pn(D4 x Z ) 36 3.3.12 Pn(Q x Z 2) 37 3.3.13 Pn(Modular-16) 3 7 3.3.14 Pn(B) 4 0 3.3.15 Pn(K) 4 1 3.3.16 Pn(G4,4) 4 2 3.3.17 Pn(Dg) 4 2

ix 3.3.18 Pn(S3 x Z 3) 46 3.3.19 P ((^ xz 3)xZ2) 4 7 3.3.20 P (DW) 4 8 3.3.21 Pn(Fr20 = Z 5 x Z 4) 4 9 3.3.22 Pn(Z 4 xz5) 5 3 3.3.23 Pn(Fr2l = Z 7 x Z 3 ) 5 7 3.3.24 Pn(D 1 1 ) 58 3.4 Conclusion 5 9 4 THE nth COMMUTATIVITY DEGREE OF 60 NONABELIAN METABELIAN GROUPS OF ORDER 24 4.1 Introduction 6 0 4.2 Some Results on Pn(G) 6 0 4.2.1 Pn (S3 x Z 4) 6 1 4.2.2 P (S x Z 2 x Z 2) 62 4.2.3 Pn(D4 x Z 3) 6 3 4.2.4 Pn(Q4 x Z 3 ) 6 4 4.2.5 Pn(A4 xz2) 6 5 4.2.6 P n((z 6 x Z 2) x Z 2) 6 5 4.2.7 Pn(D 1 2 ) 6 8 4.2.8 P (Z 2 x (Z 3 xz4)) 7 1 4.2.9 Pn(Z 3 xz8) 72 4.2.10 Pn(Z 3 xq4) 73 7 3 4.3 Conclusion 7 6 5 SUMMARY AND CONCLUSION 7 7 5.1 Introduction 77 7 7 5.2 Summary and Conclusion 77 5.3 Suggestions 78 7 8

x REFERENCES 79-81 APPENDICES 82-120 Appendix A 82 Appendix B 84 Appendix C 85 Appendix D 113

xi LIST OF TABLES TABLE NO. TITLE PAGE 3.1 Cayley Table for Z 3 x Z 4 20 3.2 0-1 Table for Z 3 x Z 20 3.3 Table for multiplication between x2 and y 21 3.4 0-1 Table multiplication between x andy 21 3.5 Table for multiplication between x andy 22 3.6 0-1 Table multiplication between x andy 23 3.7 The nth power of elements in Z 3 x Z 4 24 3.8 The nth power of elements in A 4 26 3.9 The nth power of elements in D 8 29 3.10 The nth power of elements in Quasihedral-16 31 3.11 The nth power of elements in Q8 34 3.12 The nth power of elements in Modular-16 38 3.13 The nth power of elements in B 40 3.14 The nth power of elements in D 9 44 3.15 The nth power of elements in (Z 3 x Z 3 ) x ^ 47 3.16 The nth power of elements in Fr20 = Z 5x Z 4 50 3.17 The nth power of elements in Z 4 x Z 5 54 4.1 The nth power of elements in (Z 6 x Z 2) x Z 2 6 6 4.2 The nth power of elements in D 1 2 69 4.3 The nth power of elements in Z 3x Q4 74

xii LIST OF SYMBOLS Pn(G) P(G) GAP G G, x Z Z H < G G = H G x H G' G x x Qn Dn nth commutativity degree Commutativity degree Groups, Algorithms and Programming A group G Order of the group G, the order of the element x Set of integers, the finite cyclic group Cyclic group of order n H is a subgroup of G G is isomorphic to H Direct product of G and H The commutator subgroup of G Element of Direct product Semidirect product Quarternion group of order 2n Dihedral group of order 2n

CHAPTER 1 INTRODUCTION 1.1 Introduction A group G is abelian if it satisfies the commutative law, namely xy = yx holds for every x, y e G. However, not all groups are abelian. A group that has some of the elements that do not commute is called a nonabelian group, also known as noncommutative group. Let G be a group and suppose x and y are elements of G. By considering the total number of pair (x, y) for which x and y are commute (such that xy = yx ) and divide it by the total number of pair (x, y ) which is possible, the result will give the abelianness or commutativity degree of a group G. Furthermore, by extending the notion of xy = yx to xny = yxn, the nth commutativity degree is obtained. In details, the nth commutativity degree is equal to the quotient of the total number of pair (x, y ) for which xn and y commute with the total number of pair (x, y) which is possible.

2 1.2 Research Background A metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there exists an abelian normal subgroup A such that the quotient group G/ A is abelian. In 2010, Abdul Rahman [1] determined all metabelian groups of order at most 24. There are 59 groups of order less than 24 and 15 groups of order 24 including abelian and nonabelian groups. From the results obtained, all groups of order less than 24 are metabelian. Among these 59 metabelian groups only 25 of them are nonabelian. However, among 15 groups of order 24, only two groups that are not metabelian which are symmetric group of order 24, S4 and special linear group of 2 by 2 matrices over field of three elements, SL(2,3) and the rest 13 groups are all metabelian in which 10 of them are nonabelian and three groups are abelian. The probability that two elements of the group G (chosen randomly with replacement) commute is also known as the commutativity degree of a group G and denoted as P (G ). This probability can be written as p(q^ - Number of ordered pairs (x, y) e G x G 3 xy = yx Total number of ordered pairs (x, y) e G x G _ {( x, y) e G x G xy = yx} G 2. The commutativity degree of nonabelian metabelian groups of order at most 24 has been determined by Che Mohd [2] in 2011. Her results are used throughout this research in order to find the nth commutativity degree of a group, Pn (G) where Pn (G) is defined as,

3 p ^ = {(x y) e G x G x"y = yx } n G 2 and G is a nonabelian metabelian group of order at most 24. 1.3 Problem Statement What are the nth commutativity degrees of all nonabelian metabelian groups of order at most 24? 1.4 Research Objectives The main objectives of this research are: 1) to study and present the basic concepts of metabelian groups, 2) to present the notion of commutativity degree and the nth commutativity degree of groups, 3) to determine the nth commutativity degree of nonabelian metabelian groups of order at most 24. 1.5 Scope of Research This research will focus only on nonabelian metabelian groups of order at most 24 and their nth commutativity degree.

4 1.6 Significance of Study The results obtained can be beneficial for computing the commutativity degree and the nth commutativity degree for other groups. Besides, the results of the commutativity degree and the nth commutativity degree can be transferred to noncommuting graph where this kind of graph can be used to characterize the group theory properties of a group. 1.7 Thesis Organization This dissertation is organized into five chapters. The first chapter is the introduction chapter which contains the research background, problem statement, research objectives, scope of research, significance of study, thesis organization and conclusion of the chapter. The second chapter includes some literature review about metabelian groups, commutativity degree and the nth commutativity degree of a group. Some definitions and theorems that are used throughout the research are also listed. In the following chapters, the result of nth commutativity degree for nonabelian metabelian groups of order less than 24 and equal to 24 are presented. The results are splitted into two chapters, which are Pn(G) for G is a metabelian group of order less than 24 (25 groups) and G is a metabelian group of order 24 (10 groups). Finally, the summary and conclusion of this research are included in Chapter 6.

5 1.8 Conclusion In this chapter, the introduction of this research are discussed followed by the background of the research. After that, the statement of the problem, objectives of research, scope of research and the significance of study are included. Finally, thesis organizations are stated. In the next chapter, literature reviews related to this research are discussed especially for metabelian groups, the commutativity degree and the nth commutativity degree of the groups.

79 REFERENCES 1. Abdul Rahman, S. F. Metabelian Groups of Order at Most 24. Master Dissertation. Universiti Teknologi Malaysia. 2010. 2. Che Mohd, M. The Commutativity Degree of All Nonabelian Metabelian Groups of Order at Most 24. Master Dissertation. Universiti Teknologi Malaysia. 2011. 3. Rotman, J. J. A First Course in Abstract Algebra. New Jersey. Prentice-Hall, Inc.2000. 4. Robinson, D. J. S. A Course in the Theory of Groups. New York. Springer-Verlag. 1993. 5. Redfield, R. H. Abstract Algebra A Concrete Introduction. Addison Wesley. Longman. Inc. 2001 6. Fraleigh, J. B. A First Course in Abstract Algebra. USA. Pearson Education, Inc. 2003. 7. Rotman, J. J. An Introduction to the Theory of Groups. New York. Springer-Verlag, Inc. 1994.

80 8. Erdos, P. and Turan, P. On Some Problems of Statistical Group Theory. Acta Math. Acad. O f Sci. Hung. 19:413-435. 1968. 9. Gustafson, W. H. What is the Probability that Two Group Elements Commute? American Mathematical Monthly. 1973. 80 (9): 1031-1034. 10. MacHale, D. How Commutative Can a Non-Commutative Group Be?. The Mathematical Gazette. 1974. 58:199-202. 11. Rusin, D. J. What is the Probability That Two Elements of a Finite Group Commute?. Pacific Journal o f Mathematics. 1979. 82(1):237-247. 12. Castalez, A. Commutativity Degree of Finite Groups. Master Dissertation. Wake Forest University. Winston-Salem. North Carolina. 2010. 13. Abu Sama, N., The Commutativity Degree of Alternating Groups Up To Degree 200. Undergraduate Project Report. Universiti Teknologi Malaysia. 2010. 14. Abd Hamid, M. The Probability That Two Elements Commute In Dihedral Groups. Undergraduate Project Report Universiti Teknologi Malaysia. 2010. 15. Mohd Ali, N. M. and Sarmin, N. H. On Some Problem in Group Theory of Probabilistic Nature. Menemui Matematik (Discovering Mathematics). 2010. 32(2): 35-41. 16. Yahya, Z., Mohd Ali, N. M., Sarmin, N. H., Abd Manaf, F. N. The nth Commutativity Degree of Some Dihedral Groups. Menemui Matematik (Discovering Mathematics). 2012. 34(2):7-14.

81 17. Erfanian, A., and M. Farrokhi, D. G. On the Probability of Being a 2-Engel Group. Proceeding for Biennial International Group Theory Conference (BIGTC2011). 2011. 88-90. 18. Abd Rhani. N., Generalizations on Commutativity Degree of Some Alternating Groups. Master Dissertation. Universiti Teknologi Malaysia. 2011. 19. Abd. Manaf, F. N., Sarmin, N. H., Mohd Ali, N. M., and Erfanian, A. The nth Commutativity Degree of Some Finite Groups of Nilpotency Class Two. Extended Abstract o f the 2nd Iranian Algebra Seminar. 2012. 17-20. 20. Yahya, Z., Mohd Ali, N.M., Sarmin, N.H., Johari, N. A. The nth Commutativity Degree of 2-Engel Groups of Order at Most 25. (to appear in Proceeding Simposium Kebangsaan Sains Matematik (SKSM20)). 21. Yahya. Z., Mohd Ali. N. M., Sarmin, N. H., Sabani, M. S., and Zakaria, M. The nth Commutativity Degree of Some 3-Engel Groups. (to appear in Proceeding Simposium Kebangsaan Sains Matematik (SKSM20)).