Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Algebra and Number Theory

Code

csopre1u0um17em

Title

Groups and representations (l)

Usual semester

Autumn

Published semester

2026/27/1

ECTS

3

Language

hu

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyse mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Group actions, permutation groups, automorphism groups. Semidirect products. Sylow’s Theorems. Finite p-groups. Nilpotent groups. Solvable groups, Phillip Hall’s Theorems. Free groups, presentations, group varieties. The Nielsen-Schreier Theorem. Abelian groups. The Fundamental Theorem of finitely generated Abelian groups. Torsionfree groups. Linear groups and linear representations. Semisimple modules and algebras. Irreducible representations. Characters, orthogonality relations. Induced representations, Frobenius reciprocity, Clifford’s Theorems. Transfer. Necessary prior knowledge: Lagrange’s theorem, isomorphism theorems, centralizer, normalizer, the number of elements in a conjugacy class, Cauchy’s theorem, groups given by generators and defning relations, the alternating group. diagonalizability of linear transformations

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

D.J.S. Robinson: A course in the theory of groups, Springer, 1993 I.M. Isaacs: Character theory of finite groups, Academic Press, 1976

Programmes of the course