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# Group Theory I & II: Course Outline (MAT 406) For Group Theory I

In mathematics, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups.

## Course Objective

At the end of this course the students will be able to write mathematical proofs and reason abstractly in exploring properties of groups, construct examples of and explore properties of groups, including symmetry groups, permutation groups and cyclic groups, determine subgroups and factor groups of finite groups, determine, use and apply homomorphism’s and isomorphism between groups. They would also explore the notion of group actions and Sylow’s theorem with applications.

## Course Contents

• Historical background,
• Definition of a group with some examples, Order of an element of a group, Subgroups,
• Generators and relations, Free groups, Cyclic groups, Finite groups,
• Cayley’s theorem on permutation groups,
• Cosets and Lagrange’s theorem,
• Normal subgroups, Simplicity, Normalizers, Direct products, Homomorphism,
• Factor groups, Isomorphism, Automorphism, Isomorphism theorems, Group actions,
• Stabilizers, Conjugacy classes, Sylow theorems and their applications.