MD131 Representations of groups

Faculty of Science
Autumn 2012
Extent and Intensity
3/1. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Jiří Kaďourek, CSc. (lecturer)
Guaranteed by
doc. RNDr. Jiří Kaďourek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 M5,01013
  • Timetable of Seminar Groups:
MD131/01: Wed 12:00–13:50 M6,01011, J. Kaďourek
Prerequisites
(( M1110 Linear Algebra I || M1115 Linear Algebra I ) && ( M2150 Algebra I || M2155 Algebra 1 )) || PROGRAM(N-MA) || PROGRAM(D-MA)
This subject is accessible to students having a solid knowledge of the fundamentals of the theory of groups, being acquainted with the basic notions from the theory of rings and fields, being well familiar with the foundations of linear algebra and willing to solidify and integrate their knowledge of these areas of algebra.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This course focuses on the theory of representations of finite groups by regular linear transformations of finite dimensional vector spaces. In other words, one is concerned with the study of homomorphisms of finite groups into general linear groups. The general linear group is defined to be the group of all regular linear transformations of a given finite dimensional vector space. Attention is concentrated to the case when the vector space in question is a vector space over an algebraically closed field of characteristic zero. This case encompasses the classical theory of characters of finite groups. These topics form an indispensable part of every course aimed at a deeper understanding of the theory of groups. After passing this course the students should thus be able to orient themselves safely in these parts of the theory of groups. Additionally, these areas of group theory have profound implications in the theory of finite groups. Just name the Burnside p-q theorem, and the theorem of Frobenius that until now cannot be established without invoking the character theory of finite groups.
Syllabus
  • I. Representations of groups and modules over group algebras. 1. Linear representations of groups: the definition of the concept of a representation of a group in a finite-dimensional vector space over a given field, the kernel of a representation, the degree of a representation, examples. 2. Fundamentals of the theory of modules: modules over rings, finitely generated modules, submodules, homomorphisms of modules, quotient modules, direct sums of modules, simple and semisimple modules, tensor products of modules and vector spaces. 3. Group rings and group algebras: definitions of the notions, the relationship between representations of finite groups and finitely generated modules over group algebras, equivalent representations of groups, irreducible representations of groups, direct sums of representations of groups, Schur's lemma. 4. The criterion for complete reducibility of representations of finite groups to direct sums of irreducible representations: Maschke's theorem. 5. Burnside's theorem on subgroups with finite exponent of general linear groups over fields of characteristic 0. 6. The structure of group algebras of finite groups over algebraically closed fields: direct sums of algebras over the same field, group algebras as direct sums of matrix algebras over the given field, all mutually non-isomorphic simple modules over the given group algebra, all inequivalent irreducible representations of a finite group over the given field. II. The character theory of groups. 7. Characters of groups: the definition of the concept of the character of the given linear representation of a group, characters of groups as functions of the conjugacy classes of the group, characters of the tensor products of modules over the given group algebra, the ring of virtual characters of a group. 8. Orthogonality relations: irreducible characters of groups, orthogonality relations connecting the irreducible characters of a finite group, the inner product of the functions of the conjugacy classes of a finite group, irreducible characters of a finite group over an algebraically closed field and an orthonormal basis of the vector space of all functions of the conjugacy classes of this group over the same field, the relationship between representations of a finite group over an algebraically closed field of characteristic 0 and characters of this group over the same field. 9. The character table of a finite group over an algebraically closed field: orthogonality of rows, orthogonality of columns, characters of a finite group over the field of all complex numbers. III. Applications in the theory of finite groups. 10. Induced representations of finite groups: induced modules over group algebras of finite groups, induced characters of finite groups, the Frobenius reciprocity theorem, the notion of Frobenius groups and Frobenius' theorem on these groups. Addendum. Algebraic integers: the proof of the divisibility of the order of a finite group by the degrees of the irreducible representations of this group over an algebraically closed field of characteristic 0.
Literature
  • ALPERIN, J. L. and Rowen B. BELL. Groups and representations. New York: Springer-Verlag, 1995, x, 194 s. ISBN 0-387-94525-3. info
  • ROBINSON, Derek John Scott. A course in the theory of groups. 2nd ed. New York: Springer-Verlag, 1995, xvii, 499. ISBN 0387944613. info
  • GORENSTEIN, Daniel. Finite Groups. Second edition. New York: Chelsea Publishing Co., 1980, xvii, 519. ISBN 0-8284-0301-5. info
  • DUMMIT, David Steven and Richard M. FOOTE. Abstract algebra. 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1999, xiv, 898 s. ISBN 0-13-569302-0. info
Teaching methods
Classic form of teaching consisting of lectures accompanied with seminars.
Assessment methods
The course is completed with oral examination. The accent is put on the orientation in the basic principles of the theory of representations of finite groups.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught only once.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2007, Spring 2019.
  • Enrolment Statistics (Autumn 2012, recent)
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