Prerequisite(s):None

Credit Hours : 3+0

Description and Objectives of the course

 Group theorey is the study of groups. Groups are sets eqquipped with an operation( like multiplication, adition, or composition) that satisfies certain basic properties. As the building blockes of abstract algebra, groups are so general and fundamental that they arise in nearly every branch of mathematics and the sciences. This course aimms to provide a first approach to the subject of algebra ,which is one of the basic  pillars of modern  mathematics. Thefocus of the course will be the study of groups, their types and applicatios.

Learning Outcomes

After the completion of the course, 

1. Students will have a working knowledge of important mathematical concepts in abstract algebra such as definition of a group, order of a finite group and order of an element. 
2. Students will be knowledgeable of different types of subgroups such as normal subgroups, cyclic subgroups and understand the structure and characteristics of these subgroups.
3. Students will be introduced to and have knowledge of many mathematical concepts studied in abstract mathematics such as permutation groups, factor groups and Abelian groups
4. Students will see and understand the connection and transition between previously studied mathematics and more advanced mathematics. The students will actively participate in the transition of important concepts such homomorphisms & isomorphisms from discrete mathematics to advanced abstract mathematics.
5. Students will gain experience and confidence in proving theorems

Contents 

1. Groups, Definitions and examples of groups, elementary properties of groups
2. Finite and infinite Groups
3. Order of agroup and element of a group and related results 
4. Subgroups and examplles of subgroups, subgroup tests, subgroup generated by set
5. Cyclic groups, properties of cyclic groups 
6. Classification of subgroups of cyclic groups
7. Cosets decomposition of a group,properties of cosets
8. Lagrange's theorem and its consequences. Conjugate elements and conjugacy classes
9. Centralizer, Normalizer of a subset of a group. Centre of group definition and examples
10. Normal subgroups, factor groups
11. Permutation and permutation groups,definition and examples
12.  Homomorphism of groups, Fundamental theorem of homomorphism
13. Isomorphism theorems annd Cayley's theorem
14. Endomorphism and automorphisms of groups 
15. Commutator subgroups
16. External and internal direct products,definitin and examples

Recommended Books

1. Gallian, J. A. (2017). Contemporary abstract algebra(8th ed.). New York : Brooks/Cole.
2. Mlik, D. S., Mordeson J. N.anda Sen, M. K. (1997). Fundamentals of abstract algebra. New York: WCB/McGraw-Hill Version, 10th Edition, (John Wiley and sons, 2010.)

​Suggested Books

1. Romans,S. (2021).Fundamentals of group theorey(ist ed.). Basel:Birkhauser.
2. Rose, H. E. (2006). Acourse on finite groups (ist ed. ).London:Springer-verlag
3. Fraleigh, J. B. (2003). A First course in Abstrat Algebra (7th ed. )Bostan:Addison -Wesley Publishing Company

Assessment Criteria

       Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

       Mid-Term Exam:   30

       Final-Term Exam: 50

Key Dates and Time of Class Meeting

wednesday                                              09:30 AM-11:00 AM (Reg)                      12:30 PM-02:00 PM (SS)

thursday                                                   09:30 AM-11:00 AM (Reg)                      2:00 PM-03:30 PM (SS)

Commencement of Classes                                                   October 12, 2020

Mid Term Examination                                                            December 14-18, 2020

Final Term Examination                                                          February 08-12, 2021

Declaration of Result                                                              Fenruary 19, 2021