Algebraic Structures I (MAY422)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY422
Semester	4
Course Title	Algebraic Structures I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	http://users.uoi.gr/abeligia/AlgebraicStructuresI/ASI2017/ASI2017.html

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.
General Competences	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

Activity	Semester Workload
Lectures (13x5)	65
Working independently	100
Exercises-Homeworks	22.5
Course total	187.5

Student Performance Evaluation

Final written exam in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

Attached Bibliography

J. Fraleigh: "Introduction to Algebra", Greek edition, Crete University Press, (2005).
D. Varsos, D. Deriziotis, M. Maliakas, O. Talleli, I. Emmanouil: "An Introduction to Algebra", Sofia Press, (2007).
K. Kalfa: "Introduction to Algebra", Ziti Press, (2003).
D. Poulakis: “Algebra”, Ziti Press, (2013).