Abstract Algebra (MATH-6205)

Discription

This course aims to provide a first approach to the subject of algebra, which is one of the basic pillars of modern mathematics. Abstract algebra is a broad field of mathematics, concerned with algebraic structures such as groups, rings and fields.

Learning Outcomes

After successful completion of the course, student is able to

demonstrate insight into abstract algebra with focus on axiomatic theories
apply algebraic ways of thinking
demonstrate knowledge and understanding of fundamental concepts including groups, subgroups, normal subgroups, homomorphisms and isomorphisms
demonstrate knowledge and understanding of rings, fields and their properties
understand and prove fundamental results and solve algebraic problems using appropriate techniques

Cyclic groups.
Cosets decomposition of a group.
Lagrange’s theorem and its consequences.
Conjugacy classes.
Centralizers and Normalizers.
Normal Subgroups.
Homomorphism of groups.
Cayley’s theorem.
Quotient groups.
Fundamental theorem of homomorphism.
Isomorphism theorems.
Endomorphism and automorphisms of groups.
Commutator subgroups.
Permutation groups.
p-Subgroups.
Sylow’s theorems.
Definition and examples of rings.
Special classes of rings.
Fields.
Ideals.
Ring homomorphism.

Recommended Books

Gallian, J. A. (2017). Contemporary abstract algebra (9th ed.). New York: Brooks/Cole.
Malik, D. S., Mordeson, J. N., & Sen, M. K. (1997). Fundamentals of abstract algebra. New York: WCB/McGraw-Hill.

Suggested Books

Roman, S. (2012). Fundamentals of group theory (1st ed.). Basel: Birkhäuser.
Rose, H. E. (2006). A course on finite groups (1st ed.). London: Springer-Verlag.
Fraleigh, J. B. (2003). A first course in abstract algebra (7th ed.). Boston: Addison-Wesley Publishing Company.