https://studiegids.vu.nl/en/courses/2025-2026/X_401105The student understands the structure of 푍/푛푍, its group of units (푍/푛푍)*, the Chinese Remainder Theorem, and the Euler phi-function, and can solve problems involving these concepts.The student is familiar with fundamental concepts from group theory, including subgroups, cyclic groups, generators, dihedral groups, permutation groups, centre, commutator subgroup, normal subgroups, cosets, homomorphisms, quotient groups, group actions, stabilisers, and orbits, and can apply these concepts to solve problems in concrete situations.The student knows key theorems in group theory, including Lagrange’s theorem, Cauchy’s theorem, and the First Isomorphism Theorem, and can use them to compute or prove properties in explicit cases.We study an algebraic structure called a group, which consists of a set equipped with a single binary operation satisfying certain axioms. Examples of groups include:the integers or real numbers under addition,invertible matrices (of a fixed size) under matrix multiplication,and bijections from a set to itself under function composition.Groups often appear in mathematics and science as the symmetries of objects or structures. By formalizing the common properties of these examples, we can prove general results about groups and illustrate them in various concrete cases. We will cover the following topics:Integers modulo 푛; the Chinese Remainder Theorem; Euler's phi-functionAbstract definition of a group; order of a group and of an elementExamples of groups: integers, integers modulo 푛, dihedral groups, matrix groups, etc.Subgroups, generators, homomorphismsNormal subgroups and quotient groupsCosets; index of a subgroup; Lagrange’s TheoremThe First Isomorphism TheoremCommutator subgroup; homomorphism theoremsGroup actions; orbits; the class equationCauchy’s TheoremLectures and tutorials are held for two hours each per week over 14 weeks. Study sessions take place for two hours every other week. There are four written assignments—two in each term—with the best three counting towards 5% of the final grade.For this course there will be two partial exams, four written assignments to be handed in (out of which the best three count towards the grade), as well as a resit. The grade is determined as follows: Passing the Course Through Exams The highest three scores (out of four) from the written assignments together account for 5% of the grade. The midterm exam counts for 45%. The final exam counts for 50%. Passing the Course via the Resit The rules for passing the course via the resit are the same, except that the resit score replaces the cumulative score of the midterm and final exams, contributing 95% of the grade, while homework assignments account for 5%. However, if this results in a higher grade, the resit score alone will count for 100%, and the homework scores will not be considered.David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition (2003), John Wiley and Sons.BSc Mathematics Year 1It is recommended that students are familiar with material from the VU course Basic Concepts in Mathematics, as well as relevant topics from Linear Algebra (especially matrices) and Discrete Mathematics (especially permutation groups).