MATHS 411: Abstract Algebra 1 (3)

Syllabus

 

1.  Prerequisite:  Maths 311, or permission of the department chairperson. 

 

Note:  students with strong performance in Maths 215 and Maths 217 may request permission to substitute Maths 411 for Maths 311. 

 

2.  Course Description:  The theory of groups, including subgroups, cyclic groups, normal subgroups, cosets, Lagrange’s Theorem, quotient structures, homomorphism, automorphisms, group actions, Sylow’s Theorems, structure of finite abelian groups, generators and relations.  

 

3.  Course Objectives:  The main content objective of the course is a treatment of group theory.  Groups are mathematical objects which reflect the symmetries of other objects. Examples arising from number systems, geometry, discrete mathematics, and linear algebra will already be familiar to the student.  Students completing this course will appreciate the ubiquity of symmetry and groups in mathematics.  They will see in group theory the development of a particularly simple and elegant axiomatic system.  They will gain experience in writing proofs of moderate difficulty and solving problems related to the content of the course. 

 

4.  Course Rationale:  Group theory is a major branch of modern mathematics. Moreover, it has profound connections with other branches of mathematics, including geometry, topology, analysis, and mathematical physics.  Consequently, Maths 411 is an essential course for students contemplating earning a graduate degree in pure mathematics.  For students not expecting to study at the graduate level, Maths 411 gives exposure to a particularly elegant mathematical topic.  The course primarily serves Mathematics majors, Mathematics Teaching majors, and graduate students (taught with Maths 511). 

 

5.  Course Content:  Groups and subgroups.  Cyclic groups and orders of elements.  Cosets and Lagrange’s Theorem.  Normal subgroups, quotient structures, homomorphisms, isomorphisms, and automorphisms.  Conjugacy classes, centralizers, and normalizers.  Introduction to group actions.  Sylow’s Theorems.  The structure of finite abelian groups.  Generators and relations.  Examples coming from number systems, symmetric and alternating groups, symmetry groups of geometric objects, matrix groups, and direct products.  Additional topics at the discretion of the instructor.

 

6.  Course Format: lecture/discussion. 

 

7.  Methods of Evaluating Student Performance: Typical components are homework assignments, projects, and take-home exams (possibly involving group work); in-class exams; student presentations and oral exams.  The evaluation and weight of these components are at the discretion of the instructor.  Students may be evaluated on their understanding of the content material of the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others orally and/or in writing.  

 

8.  Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.  

 

R. Bremigan, J. Lorch, 11/2000

V. Mascioni, 4/2004

M. Karls Spring 2006