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The Relative Commutativity Degree and Sub-Multiplicative Degree for Noncyclic Subgroups of Some Nonabelian Metabelian Groups

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The Relative Commutativity Degree and Sub-Multiplicative Degree for Noncyclic Subgroups of Some Nonabelian Metabelian Groups 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 ACKNOWLEDGEMENTS 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 Ministry of Higher Education (MOHE) Malaysia for sponsoring my master study through MyBrain’ 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 ABSTRACT 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 ABSTRAK 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 danMaple, 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 TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION ii DEDICATION iii ACKNOWLEDGEMENTS iv A BSTRACT v A BSTRA K vi TABLE OF CONTENTS vii LIST OF TABLES xi LIST OF SYMBOLS xii 1 INTRODUCTION 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 LITERATURE REVIEW 9 2.1 Introduction 9 2.2 Metabelian Groups 9 2.3 The Commutativity Degree of a Group 12 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 NONCYCLIC SUBGROUPS OF SOME NONABELIAN METABELIAN GROUPS 19 3.1 Introduction 19 3.2 Steps to Find Noncyclic Subgroups by Using GAP Software 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 A4 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 xZ2 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 G4 ,4 27 3.3.10 Noncyclic Subgroups of S3 x Z 3 27 3.3.11 Noncyclic Subgroups of (Z3 x Z 3 ) x Z 2 28 3.3.12 Noncyclic Subgroup of Z5 x Z 4 28 Noncyclic Subgroups of Dihedral Groups of Order at Most 24 29 3.4.1 Noncyclic Subgroups of D4 29 3.4.2 Noncyclic Subgroups of D6 30 ix 3.4.3 Noncyclic Subgroups of D 8 30 3.4.4 Noncyclic Subgroups of D 9 31 3.4.5 Noncyclic Subgroups of Di0 31 3.4.6 Noncyclic Subgroups of D12 31 3.5 Conclusion 33 4 RELATIVE COMMUTATIVITY DEGREE OF SOME NONABELIAN METABELIAN GROUPS 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-MULTIPLICATIVE DEGREE OF SOME NONABELIAN METABELIAN GROUPS 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 CONCLUSION 111 6 . 1 Introduction 1 1 1 6.2 Summary 111 6.3 Suggestions for Further Research 114 REFERENCES 114 A PPEN D ICES 117 x A Publication and Presentation 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 Nonabelian Metabelian Groups of Order Less Than 24 120 C Cayley Table of Dihedral Groups of Order at Most 24 131 xi LIST OF TABLES TABLE NO. TITLE PAGE 2 . 1 The nonabelian metabelian groups of order less than 24 1 0 2 . 2 Dihedral groups of order at most 24 1 1 2.3 The nonabelian metabelian groups of order less than 24 that have noncyclic subgroups 1 1 2.4 Dihedral groups of order at most 24 that have noncyclic subgroups 1 2 3.1 Number of noncyclic subgroups of nonabelian metabelian groups of order less than 24 29 3.2 Number of noncyclic subgroups of dihedral groups of order at most 24 32 4.1 The Cayley table of 4A 36 4.2 The 0-1 Table of A 4 36 4.3 The 0-1 Table for multiplication between elements in {e, a, b, ab] and elements of A4 37 6 . 1 Relative commutativity degree for noncyclic subgroups of nonabelian metabelian groups of order less than 24 1 1 2 6 . 2 Relative commutativity degree for dihedral groups of order at most 24 113 6.3 The sub-multiplicative degree for noncyclic subgroups of some nonabelian metabelian groups 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 andg 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 INTRODUCTION 1.1 Introduction 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 Dn/N is a cyclic group of order two, hence abelian.
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