Objectives and competences
Learning fundamental algebraic concepts and
abstract thinking.
Content (Syllabus outline)
An overview of algebraic structures:
semigroups, groups, rings, fields, vector spaces,
algebras. Substructures. Generators. Direct
products and sums.
Examples of groups and rings: the integers, the
integers modulo n, the quaternions, matrix rings
and linear groups, rings of functions, polynomial
rings, symmetric groups, dihedral groups.
Homomorphisms: basic notions and examples.
Cayley's theorem. Field of fractions.
Quotient structures: normal subgroups and
quotient groups, ideals and quotient rings,
isomorphism theorems.
Finite groups: Lagrange's theorem, Caucy's
theorem, group actions, Sylow theorems,
simple groups, finite Abelian groups.
Learning and teaching methods
- Lectures
- Tutorial
Intended learning outcomes - knowledge and understanding
Knowledge and Understanding:
- The knowledge of basic algebraic
structures and their substructures,
homomorphisms, and quotient
structures.
- Understanding the basics of the theory
of finite groups.
Intended learning outcomes - transferable/key skills and other attributes
Transferable/Key Skills and other attributes:
- The obtained knowledge is a
prerequisite for a study of almost any
area of mathematics.
Readings
M. Brešar, Uvod v algebro, DMFA, 2018.
M. Brešar, Undergraduate algebra. A unified approach, Springer, 2019.
D. S. Dummit, R. M. Foote, Abstract Algebra, Prentice-Hall International, Inc., 1991.
J. Gallian: Contemporary Abstract Algebra, Brooks/Cole, 2013.
I. Vidav, Algebra, DMFA, 1980.
Prerequisits
Linear algebra
Additional information on implementation and assessment Type (examination, oral, coursework, project):
Written exam – problems
Oral exam – theoretical part
Written exam can be replaced by two partial tests (mid-term testing).
Each of the two exams, oral and written, must be assessed with a passing grade.
Passing the written exam is a prerequisite for taking the oral exam.
