Math445

Course Description form

Faculties and departments requiring this course (if any): Faculty of

          Science/Department of Mathematics.

Objectives: 

1- To teach areas in algebra those are not covered in the other

     algebra courses.

2- To develop the students abstract and logical thinking capabilities.

3- To develop the students mathematical ability to handle proofs in

     finite group theory.

Contents:

      I- Series

·         Normal series

·         Composition series

·         Invariant series.

·         Chief series

·         Jordan-Holder theorem

II- Characteristic subgroups and commutators   

·         Automorphism

·         Characteristic subgroups

·         Minimal normal subgroups (definition and examples)

·         Characteristically simple groups (definition and examples)

·         Commutators (definition and examples) 

·         Commutators group (derived group)

III- Solvable groups 

·                     Subnormal subgroups (definition, examples and theorems)

·                     Solvable groups (definitions and examples)

·                     Mean theorems on solvable groups

III- Nilpotent groups

·         Lower and upper central series

·         Nilpotent groups

·         Mean theorems on nilpotent groups

Course Outcomes:

A-   Knowledge:

    (Specific facts and knowledge of concepts, theories, formula etc.)

    On completion of this course, students should be recognize that finite group theory is one of the famous branches in abstract algebra which play a central role in various disciplines of physics like mechanics, electronics and acoustics. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure preserving transformations. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the theory of solvable groups and nilpotent groups.

B-Cognitive Skills:

     (Thinking, problem solving)

     1- Applying the standard emanating criteria given in the course to much better understanding applied models.

     2- Recognizing the importance of group theory as wonderful and power tools to better understand what the students study.

C- Interpersonal skills and responsibilities:

1.    Questioning during lecture.

2.    Submitting assignments.

3.    Group discussions.

4.    Recognizing the importance of the team work.

D- Analysis and communication:

     (communication, mathematical and IT skills)

     The students must recognize that group theory is a core subject in mathematics, more precisely in abstract algebra and it has many applications.  For example, this course helps the student to understand what was behind the various reduction techniques in differential equations, which facilitate their solutions.  

     Assessment methods for the above elements: Quizzes, assignments, and examinations.

Text book:

     A Course in group Theory (1996)

     Author: John F.  Humphreys

     Publisher: Oxford University Press

Supplementary references

     Finite group Theory, 2ed (2000).     

    Author: M. Aschbacher 

    Publisher: Cambridge University Press.