Introduction to Abstract Algebra

Revised: November 2006

Course Description

Groups, rings, and fields. Prerequisite: Math 250. Three semester hours.

Objectives

   1. To provide a knowledge of basic algebraic structures (groups, rings and fields).

   2. To provide practice is proof writing techniques.

   3. To prepare students for higher level mathematics courses.

Text

Joseph A. Gallian, Contemporary Abstract Algebra, Fifth Edition. Houghton Mifflin.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• Chapter 0: Preliminaries (3 days)
        Properties of integers, modular arithmetic, mathematical induction, equivalence relations, Functions (mappings)
• Chapter 1: Introduction to Groups (2 days)
        Symmetries, dihedral groups
• Chapter 2: Groups (3 days)
        Definitions, examples, elementary properties
• Chapter 3: Finite Groups: Subgroups (3 days)
        Notation, subgroup tests, examples of subgroups
• Chapter 4: Cyclic Groups (3 days)
        Properties of cyclic groups, classification of cyclic subgroups
• Chapter 5: Permutation Groups (4 days)
        Definition and notation, properties of permutations
• Chapter 6: Isomorphisms (3 days)
        Definition and examples, Cayley's Theorem, properties of isomorphisms, automorphisms
• Chapter 7: Cosets and Lagrange's Theorem (3 days)
        Properties of cosets, Lagrange's Theorem and consequences, applications
• Chapter 8: External Direct Products (3 days)
        Definition and examples, properties of External Direct Products, applications
• Chapter 9: Normal Subgroups and Factor Groups (4 days)
        Normal subgroups, factor groups, applications, internal direct products
• Chapter 10: Group Homomorphisms (3 days)
        Definition and examples, properties, the First Isomorphism Theorem
• Chapter 11: Fundamental Theorem of Finite Abelian Groups (3 days)
        The Fundamental Theorem, Isomorphism classes of Abelian groups, Proof of the Fundamental Theorem
• Chapter 12: Rings (3 days)
        Definition and examples, properties, subrings
• Chapter 13: Integral Domains (2 days)
        Definition and examples, properties, fields, characteristic of a ring

    * Note: At appropriate places in this course, time should be allotted to elaborate on the historical aspects relevant to the subject.