# Algebraic Structures I (MAY422)

### General

School School of Science Department of Mathematics Undergraduate MAY422 4 Algebraic Structures I Lectures (Weekly Teaching Hours: 5, Credits: 7.5) General Background - Greek, English Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The course aims to introduce the students to the study algebraic properties of sets which are equipped with one or more (binary) operations. Such mathematical objects are called algebraic structures. We will mainly deal with two types of algebraic structures: Groups. The standard example is the group of permutations of a, usually finite, set. This is the set of all bijective functions from a set to itself endowed with the operation of composition of functions. Rings. The standard example of a ring is the set of integers equipped with the operations of addition and multiplication of integers. We will formulate various theorems concerning the structure and basic properties of groups and rings emphasizing the concept of isomorphism of groups or rings. From the perspective of Algebra two algebraic structures which are isomorphic, they have exactly the same algebraic properties. As a direct consequence, results concerning an algebraic structure are valid in any isomorphic algebraic structure. In the course we present several examples illuminating various notions of symmetry. It should be noted that the notion of symmetry is the central theme which underlies the concept of group/ring. At the end of the course we expect the student: (a) to have understood the definitions and basic theorems which are discussed in the course, (b) to have understood how they are applied in discrete examples, (c) to be able to apply the material in order to extract new elementary conclusions, and finally (d) to perform some (no so obvious) calculations. The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of algebraic structures, in particular of the general theory of Groups and Rings, which form an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of groups and rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

### Syllabus

• Preliminaries: Sets, functions, equivalence relations, partitions, (binary) operations.
• Groups – Permutation groups.
• Cyclic groups – generators.
• Cosets with respect to a subgroup – Lagrange’s Theorem.
• Homomorphisms of groups – Quotient groups.
• Rings and fields - Integral domains.
• The theorems of Fermat and Euler.
• Polynomial rings – Homomorphisms of Rings.
• Quotient rings – Prime and maximal ideals.

### Teaching and Learning Methods - Evaluation

Delivery

Classroom (face to face)

Use of Information and Communications Technology

Teaching Material: Teaching material in electronic form available at the home page of the course.
Communication with the students:

• Office hours for the students (questions and problem solving).
• Email correspondence
• Weekly updates of the homepage of the course.
Teaching Methods