PMAT 42433: Group Theory

Course Code     : PMAT 42433

Title                   : Group Theory

Pre-requisites    : PMAT 21263

 Learning Outcomes:

After the completion of this course unit, the student will be able to:

1. analyze and use basic definitions in Abstract algebra including binary operations, groups, subgroups, homomorphism, group actions and Sylow theorems

2. examine the fundamental principles including the laws and theorems arising from the concepts covered in this course

3. develop and apply the fundamental properties of abstract algebraic structures, the substructures, their quotient structures and their mappings

4. build experience and confidence in proving theorems about the structure size and nature of groups, subgroups, group Isomorphism, group actions, Sylow theorems and the associated mappings

5. apply course materials along with techniques and procedures covered in this course to solve problems

6. develop specific skills, competencies and thought processes sufficient to support further studies or work in this or related fields

7. demonstrate knowledge of the structure of Groups and to apply the knowledge in solving problems in different areas in Algebra.

 Course Contents:

Groups: 

• Definition and Examples, Basic properties, Subgroups, symmetric Groups, Dihedral Groups, Cyclic Groups, Abelian Groups, Finite & Infinite Groups, order of a group, order of an element, Structure theory of finite Abelian Groups.

Normal subgroups: 

• Definition and examples, Quotient groups, Cosets, Lagrange theorem, Internal direct product.

Group isomorphism: 

• Definition of Group Homomorphism, Kernel of a homomorphism, Image of a homomorphism, Group Isomorphism, Isomorphism Theorems (First, Second and Third Isomorphism Theorems)

Group action: 

• Orbits and Stabilizers, Orbit Stabilizer theorem, Burnside's theorem, Fixed point convergence theorems, Cauchy’s theorem, Fermat’s Little theorem.

Sylow Theorems: 

• Sylow’s p- subgroups, First, Second and Third Sylow’s Theorems, Applications of the Sylow’s Theorem, Groups of small order.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

 Assessment : Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading: 

1. Khanna, V.K. & Bhambri, S.K. (2016). A Course in Abstract Algebra, Vikas Publishing House.

2. Frakeigh, J.B. (2003). A first course in Abstract Algebra, Pearson Education India.

3. Baumslag, B. & Chandler, B. (1968). Group theory, McGraw-Hill, New York.

4. Narayan, S. & Pal, S. (1992). A Textbook of Modern Abstract Algebra, S. Chands, India.

5. Rotman, J.J. (4th Ed., 2014). An Introduction to the Theory of Groups, Springer-Verlag.

6. Gilbert, L. (8th Ed., 2014). Elements of Modern Algebra, Cengage Learning.