Introductory Group Theory

Course Description: This course will serve as an introduction to abstract algebra by going over the fundamentals of group theory. The course will study how algebra can arise from concepts such as symmetry and rotation. Basic examples of groups like permutation groups and dihedral groups will be examined. After this, some applications of group theory in areas such as twisty puzzles (e.g. the Rubik’s Cube) will be discussed. The course will end with an exam.

Length: 6 Weeks

Goals:

• Understand the reasons behind abstraction in mathematics

• Understand the basics of group theory and abstract algebra

• Outline dihedral groups and permutation groups

• Be familiar with the notation

• Solve example problems related to dihedral groups and permutation groups

• Understand how abstract algebra concepts can be applied to the Rubik’s Cube, the Pyraminx, and other twisty puzzles

Course Outline:

Week 1: The History of Abstract Algebra and Group Axioms

Week 2: Definition of Function and Permutation Groups

Week 3: Cycles and Cyclic Groups, Dihedral Groups

Week 4: Isomorphisms, Homomorphisms, Subgroups, and Related Concepts

Week 5: Problem Solving

Week 6: Application of Abstract Algebra Concepts on the Rubik’s Cube