### Algebra I (MATH 511)

2022 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10
Michel Lavrauw mlavrauw@sabanciuniv.edu,
English
Doctoral, Master
--
Formal lecture
Discussion based learning

### CONTENT

Introduction to group theory. Isomorphism theorems. Permutation groups and Cayley's theorem. Conjugacy classes. Lagrange's theorem and the Sylow theorems. principle ideal domains. Polynomial ring.

### OBJECTIVE

This is the first part of the two-semester basic algebra course for graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures include groups, rings, vector spaces, modules and fields.

### LEARNING OUTCOMES

• Define and use Subgroups, Ideals, Homomorphisms and other basic concepts about Groups and Rings,
• Analyze and produce proofs of statements involving Groups and Rings,
• Use some fundamental examples such as the Symmetric Group and its Subgroups, Cyclic Groups, Matrix Groups, and Polynomial Rings,
• Distinguish basic Algebraic structures from each other up to Isomorphism,
• Use the three Isomorphism theorems for Groups and Rings,
• Use Direct and Semi-direct Product Constructions,
• Use Group Actions to study structures of Groups and to do effective counting in Groups,
• Apply Sylow Theorem and the Structure Theorem on Finitely Generated Abelian Groups,
• State basic Ring types such as Local Rings, Principal Ideal Domains, Unique Factorization Domains.

### ASSESSMENT METHODS and CRITERIA

 Percentage (%) Final 50 Midterm 30 Participation 10 Homework 10