Theory of semigroups and groups

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

I. Semigroup theory
basic notions and examples
Green relations
Regular semigroups, inverse semigroups.
Simple semigroups, completely simple semigroups.
II. Group theory
Basic notions
Composition series, Jordan-Hölder theorem. Solvable groups, Hall's theorem. Nilpotent groups, p-groups.
Split and non-split extensions of groups, semidirect product, Schur-Zassenhaus theorem.
Finite simple groups and the classification problem. Classical groups (general linear, symplectic, unitary and orthogonal) and the corresponding simple groups.
Fundamentals of representation theory of finite groups. Character theory.

Readings

J. M. Howie: Fundamentals of semigroup theory, Oxford University Press, Oxford, 1995.
P. M. Higgins: Techniques of semigroup theory, Oxford University Press, Oxford, 1992.
J. J. Rotman: An introduction to the theory of groups, 4. izd., Springer New York 1995.
D. J. S. Robinson: A course in the theory of groups, 2. izd., Springer New York, 1996.

Objectives and competences

Students get acquainted with basic notions of group theory and semigroup theory. They get familiar with connections between these two theories and other areas of mathematics.

Intended learning outcomes

Knowledge and understanding:
Basic notions of group theory and semigroup theory, applications in other areas of mathematics.
Application:
Group theory and semigroup theory are classical mathematical disciplines. They teach us how to recognize symmetries. They have immense applications in physics and chemistry (crystallography). Within mathematics, they play an important role in geometry, associative algebra, functional analysis, and number theory.
Reflection:
Understanding theory based on examples and applications.
Transferable skills:
Formulation of problems, solving problems and analysis of results using examples, applying groups in geometry and analysis.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Jaka Cimprič:
CIMPRIČ, Jaka. Real spectra of quantum groups. Journal of algebra, ISSN 0021-8693, 2004, vol. 277, no. 1, str. 282-297. [COBISS-SI-ID 13108569]
CIMPRIČ, Jaka. Preorderings on semigroups and semirings of right quotients. Semigroup forum, ISSN 0037-1912, 2000, vol. 60, no. 3, str. 396-404. [COBISS-SI-ID 9568857]
CIMPRIČ, Jaka. On homomorphisms from semigroups onto cyclic groups. Semigroup forum, ISSN 0037-1912, 1999, let. 59, št. 2, str. 183-189. [COBISS-SI-ID 8951641]
Tomaž Košir:
KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. Maximal semigroups dominated by 0-1 matrices. Semigroup forum, ISSN 0037-1912, 1997, let. 54, št. 2, str. 175-189. [COBISS-SI-ID 7306329]
GRUNENFELDER, Luzius, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. On groups generated by elements of prime order. Geometriae dedicata, ISSN 0046-5755, 1999, let. 75, št. 3, str. 317-332. [COBISS-SI-ID 8849241]
BERNIK, Janez, DRNOVŠEK, Roman, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. Irreducible semigroups of matrices with eigenvalue one. Semigroup forum, ISSN 0037-1912, 2003, vol. 67, no. 2, str. 271-287. [COBISS-SI-ID 12583257]
Primož Moravec:
MORAVEC, Primož. Unramified Brauer groups of finite and infinite groups. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 6, str. 1679-1704. [COBISS-SI-ID 16521305]
MORAVEC, Primož. On the Schur multipliers of finite p-groups of given coclass. Israel journal of mathematics, ISSN 0021-2172, 2011, vol. 185, no. 1, str. 189-205. [COBISS-SI-ID 16311129]
MORAVEC, Primož. Completely simple semigroups with nilpotent structure groups. Semigroup forum, ISSN 0037-1912, 2008, vol. 77, no. 2, str. 316-324. [COBISS-SI-ID 14768473]
Primož Potočnik:
POTOČNIK, Primož. Edge-colourings of cubic graphs admitting a solvable vertex-transitive group of automorphisms. Journal of combinatorial theory. Series B, ISSN 0095-8956, 2004, vol. 91, no. 2, str. 289-300. [COBISS-SI-ID 13087321]
MALNIČ, Aleksander, MARUŠIČ, Dragan, POTOČNIK, Primož. On cubic graphs admitting an edge-transitive solvable group. Journal of algebraic combinatorics, ISSN 0925-9899, 2004, vol. 20, no. 1, str. 99-113. [COBISS-SI-ID 13267033]
POTOČNIK, Primož. B-groups of order a product of two distinct primes. Mathematica slovaca, ISSN 0139-9918, 2001, vol. 51, no. 1, str. 63-67. [COBISS-SI-ID 10617433]