The homological functors are originated in homotopy theory. One of the homological functors, namely the nonabelian tensor square, was introduced by Brown and Loday in 1987. Meanwhile, the rest of the homological functors were introduced by Bacon and Kappe in 2003. Some of the homological functors are J (G), �(G), exterior square, the Schur multiplier, �(G), the symmetric square and J�(G). The nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two are determined in this research. A software named Groups, Algorithms and Programming (GAP) is used to compute and verify the nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two

Zainal, Rosita

Universiti Teknologi Malaysia Institutional Repository

THE HOMOLOGICAL FUNCTORS OF SOME FINITE TWO-GENERATOR  TWO-GROUPS OF NILPOTENCY CLASS TWO           ROSITA BINTI ZAINAL             UNIVERSITI TEKNOLOGI MALAYSIA THE HOMOLOGICAL FUNCTORS OF SOME FINITE TWO-GENERATOR  TWO-GROUPS OF NILPOTENCY CLASS TWO     ROSITA BINTI ZAINAL     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 2012 iii                To my beloved mother, father and family                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. Besides, I would like to thank also Assoc. Prof. Dr. Nor Haniza Sarmin for her comments and suggestions to improve my research.  Moreover, I would like to express my thanks to my parents, Hajjah Rokiah Binti Abdullah and Haji Zainal Bin Haji Sulong for their long lasting love. Lastly, I would like to thank my siblings and my friends. 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     The homological functors are originated in homotopy theory. One of the homological functors, namely the nonabelian tensor square, was introduced by Brown and Loday in 1987. Meanwhile, the rest of the homological functors were introduced by Bacon and Kappe in 2003. Some of the homological functors are J (G), ( )G , exterior square, the Schur multiplier, ( )G , the symmetric square and ( )J G .  The nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two are determined in this research. A software named Groups, Algorithms and Programming (GAP) is used to compute and verify the nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two.             vi  ABSTRAK     Fungtor homologi adalah berasal daripada teori homotopi. Salah satu fungtor homologi adalah kuasa dua tensor tak abelan telah diperkenalkan oleh Brown dan Loday pada tahun 1987. Sementara itu, fungtor homologi yang lainnya telah diperkenalkan oleh Bacon dan Kappe pada tahun 2003. Di antara fungtor homologi adalah  J (G), ( )G , kuasa dua peluaran, pendarab Schur, ( )G , kuasa dua simetrik dan ( )J G .  Kuasa dua tensor tak abelan dan beberapa fungtor homologi untuk sebahagian kumpulan-dua dengan penjana-dua yang mempunyai nilpoten kelas dua terhingga telah ditentukan dalam penyelidikan ini. Perisian “Groups, Algorithms and Programming” (GAP) telah digunakan untuk mengira dan menentusahkan kuasa dua tensor tak abelan dan beberapa fungtor homologi untuk sebahagian kumpulan-dua dengan penjana-dua yang mempunyai nilpoten kelas dua terhingga.               vii  TABLE OF CONTENTS     CHAPTER    TITLE         PAGE    DECLARATION               ii   DEDICATION               iii   ACKNOWLEDGEMENTS                         iv   ABSTRACT                v   ABSTRAK                vi   TABLE OF CONTENTS              vii   LIST OF TABLES               x   LIST OF SYMBOLS              xi   1 INTRODUCTION               1   1.1 Introduction              1   1.2 Research Background             2   1.3 Problem Statement             2   1.4 Research Objectives             2 1.5 Scope of Research             3 1.6 Significance of Study             3 1.7 Research Methodology            4 1.8 Dissertation Report Organization           4 viii   2 LITERATURE REVIEW         6   2.1 Introduction          6   2.2 Some Basic Concepts in Group Theory      8 2.3 Some Basic Definitions and Concepts on Homological      Functors                                                                                    9 2.4 The Classification of Two-Generator Two-Groups of     Nilpotency Class Two      12   2.5 Groups, Algorithms and Programming (GAP) Software  on Computing The Nonabelian Tensor Square and Some  Homological Functors      13   2.6 Conclusion        14  3 THE HOMOLOGICAL FUNCTORS OF SOME FINITE TWO-GENERATOR TWO-GROUPS OF NILPOTENCY  CLASS TWO OF TYPE 1      15   3.1 Introduction       15   3.2 Some Preparatory Results     16   3.3 Some Homological Functors of Some Finite Two-Generator  Two-Groups of Nilpotency Class Two of Type 1              17   3.4 GAP Verification      21   3.5 Conclusion       25  4 GAP COMPUTATION OF SOME HOMOLOGICAL FUNCTORS OF FINITE TWO-GENERATOR TWO- GROUPS OF NILPOTENCY CLASS TWO OF TYPE 3           26   4.1 Introduction                  26   4.2 GAP Computation of Type 3     27 4.3 Conclusion                    38   5 SUMMARY AND SUGGESTIONS               39   5.1 Summary                                           39             5.2 Suggestions                  40                   ix    REFERENCES                  41   Appendices            44 - 64                                   x   LIST OF TABLES     TABLE NO.    TITLE         PAGE  3.1 The nonabelian tensor square, ( )J G  , exterior square and   Schur multiplier for group of Type 1 with                 24  4.1  The nonabelian tensor square, ( )J G , exterior square and  Schur multiplier for groups of Type 3 with   ,   	2  and 5       	                                                                    30   4.2  The nonabelian tensor square, ( )J G , exterior square and  Schur multiplier for groups of Type 3 with ,   	2  and 5     	  	                                                                    32   4.3  The nonabelian tensor square, ( )J G , exterior square and  Schur multiplier for groups of Type 3 with ,  	 	2  and 5       	                                                                    35   4.4  The nonabelian tensor square, ( )J G , exterior square and  Schur multiplier for groups of Type 3 with   ,  	 	2  and 5     	  	                                                                    37         xi   LIST OF SYMBOLS     1  Identity element 1  Identity of tensor square GAP  Groups, Algorithms and Programming G  A group G ,G x   Order of the group G, the order of the element x    Set of integers, the finite cyclic group  n   Cyclic group of order n  x   Group generated by x  ,g h   The commutator of g and h H G   H is a subgroup of G G H   G is isomorphic to H G H   Direct product of G and H G H  Tensor product of G and H G G   Tensor square of G G G   Symmetric square of G G G   Exterior square of G  M G   Schur multiplier of G  Z G   Center of the group G  ker   Kernel of the homomorphism   g h   The conjugate of h by g G   The commutator subgroup of G    Subset   Element of   Wedge product   Direct product xii  	  Greater than   Less than   Greater than or equal   Less than or equal   Semidirect product CHAPTER 1     INTRODUCTION     1.1 Introduction   The homological functors of a group including  J G ,  G , the exterior square, the Schur multiplier,  G , the symmetric square and  J G  are closely related to the nonabelian tensor square of the group.  The nonabelian tensor square of a group G, denoted as G G , is generated by g h  for all ,g h G  subject to the relations   g ggg h g h g h      and   .h hg hh g h g h      Throughout the years, many researchers have been conducted to investigate the nonabelian tensor square and homological functors of various groups.   This research begins on viewing the history of the nonabelian tensor square and the homological functors for some groups that have been done by previous researchers. In this research, we will focus on some finite two-generator two-groups of nilpotency class two. The nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two have been determined in this research. Groups, Algorithms, and Programming (GAP) software, is used to compute and verify the results.   2  1.2 Research Background   The classification of finite two-generator two-groups of nilpotency class two and  their  nonabelian  tensor  square  have  been  determined  by  Kappe et al. [1] in  1999. In [2], by adding a new parameter, the classification for Type 2.5.3 has been corrected. Therefore, the nonabelian tensor square for Type 2.5.3 given in [1] is not valid.  The new nonabelian tensor square together with some homological functors of some finite two-generator two-groups of nilpotency class two of those types (later will be called Type 3) are computed in this research. In [2], some homological functors of finite two-generator two-groups of nilpotency class two of Type 2.4 (later will be called Type 1) are determined. Extending the research done in [1], some homological functors of finite two-generator two-groups of nilpotency class two of Type 1 for parameter      have been determined in this research by following the definition of homological functors given in [3]. The results are then verified using Groups, Algorithms, and Programming (GAP) software. The GAP software is also used for the computation of some homological functors of some finite two-generator two-groups of nilpotency class two.   1.3 Problem Statement   What are the nonabelian tensor square for finite two-generator two-groups of nilpotency class two of Type 3 and what are the homological functors of some homological functors for finite two-generator two-groups of nilpotency class two of Type 1 and Type 3?   1.4 Research Objectives   The main objectives of this research are: 3  1. to study the classification of finite two-generator two-groups of           nilpotency class two,  2. to determine the homological functors of some finite two-generator two-groups of nilpotency class two for Type 1, 3. to compute the nonabelian tensor square and some homological functors of some finite two-generator two-groups of nilpotency class two for Type 3, 4. to verify all the results by Groups, Algorithms and Programming (GAP) software.   1.5 Scope of Research   This research focuses only on Type 1 and Type 3 in the classification of finite two-generator two-groups of nilpotency class two. The homological functors computed include only the nonabelian tensor square, J(G), exterior square, the Schur multiplier, and ( )J G .   1.6 Significance of Research   This research will lead to the development of new theorems with proofs. The results of this research can be presented in a conference and can be sent for publication in a journal. Furthermore, the results of this research can be used for further research that related to two-generator two-groups of nilpotency class two and related research in homological functors. This research also enhances contribution from mathematicians in Malaysia especially in Pure Mathematics fields.     4  1.7 Research Methodology   This research has been carried out according to the following steps: 1. Study the classification of finite two-generator two-groups of nilpotency class two. 2. Study on nonabelian tensor square of finite two-generator two-groups of nilpotency class two. 3. Study on the homological functors of finite two-generator two-groups of nilpotency class two of Type 1, Type 2, and Type 4. 4. Compute some homological functors of some finite two-generator two-groups of nilpotency class two of Type 1. 5. Compute the nonabelian tensor square and some homological functors of finite two-generator two-groups of nilpotency class two of Type 3.  6. Write up of dissertation. 7. Presentation of dissertation.   1.8 Dissertation Report Organization    This dissertation is organized into five chapters. Chapter 1 is the research framework. This chapter includes the research background, problem statement, research objectives, scope of research, significance of study, and the research methodology.       In Chapter 2, some definitions and basic concepts that will be used throughout the dissertation are included. Among these are some basic concepts in group theory and homological functors and also the classification of two-generator two-groups of nilpotency class two. Some introduction of Groups, Algorithms and Programming (GAP) software is also included in this chapter.   5  Chapter 3 reviews on some results obtained by previous researchers. By using the complete classification of finite two-generator two-groups of nilpotency class two, the homological functors of some finite two-generator two-groups of nilpotency class two of Type 1 are presented. These include the proof for each homological functors calculated. The results are then verified by using Groups, Algorithms and Programming (GAP) software.   Chapter 4 discusses the use of Groups, Algorithms and Programming (GAP) software in the computation of the results. Some GAP programmes that generate the general code of the nonabelian tensor square and the homological functors of some finite two-generator two-groups of nilpotency class two of Type 3 are provided in this chapter. The general formula of the nonabelian tensor square and some homological functors for some cases are also included.    Lastly, in Chapter 5, the obtaining results are summarized. Suggestions for further research are also included. 41       REFERENCES    1. Kappe,  L.C.,   Sarmin,  N.H.   and   Visscher, M.P.  Two-Generator   Two-Groups  of        Class  Two  and  Their  Nonabelian  Tensor  Squares. Glasgow Math. J. 1999. 41:417-     430.  2. Semil   @   Ismail,  G.   The  Homological  Functors  of  Some  Two-Generator  Two-     Groups   of   Nilpotency   Class   Two.   Master   Dissertation.  Universiti    Teknologi      Malaysia. 2009.   3. Bacon, M.   and   Kappe, L.C.  The  Nonabelian  Tensor  Square  of  a  2-Generator p-       Group of Class 2. Arch. 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The homological functors of some finite two-generator two-groups of nilpotency class two

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The homological functors of some finite two-generator two-groups of nilpotency class two

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