COMPETENCES



M11CM01 - To understand the concept of action of a group on a set and the equivalent concept of permutation representation.

M11CM02 - To know Sylow's theorems and to be able to apply them in order to prove the solubility of some groups and to classify groups of small order.

M11CM03 - To understand the equivalence between the concept of group representation and that of action on a vector space.

M11CM04 - To know how to define some basic group representations.

M11CM05 - To understand Maschke's theorem and its role in representation theory.

M11CM06 - To know the concept of a character and its main properties.

M11CM07 - To know how to calculate the character table of a group in some easy cases.

M11CM08 - To understand Burnside's theorem showing the solubility of groups of order p^aq^b.



LEARNING RESULTS



- To know the concepts and applications regarding actions of groups on sets.

- To know Sylow's theorems and its applications (classification of groups of small order and criteria for non-simplicity).

- To know how to define some basic group representations.

- To know how to calculate the character table of a group in some easy cases.

1. FREE GROUPS AND GROUP PRESENTATIONS: Free groups. Universal property of free groups. Group presentations. Von Dyck's theorem. Examples.

2. GROUP ACTIONS ON SETS: Actions and permutation representations. Orbits and stabilizers. Conjugacy classes and centralizers. Actions of groups on groups and semidirect product.

3. SYLOW'S THEOREMS: Sylow subgroups. Sylow's theorems. Applications: criteria for non-simplicity and classification of some groups of small order.

4. SOLUBLE GROUPS: Commutators of elements and commutators of subgroups. The derived subgroup and the derived series. Soluble groups. Minimal normal subgroups in finite soluble groups.

5. GROUP REPRESENTATIONS: The concept of representation. Group representations. Irreducible representations and Schur's lemma. Maschke's theorem.

6. CHARACTERS: Character of a representation. Properties. Orthogonality relations. The space of class functions. Kernel and centre of a character.

7. BURNSIDE'S p^aq^b THEOREM: Algebraic integers. Divisibility of the degrees of the irreducible characters. Burnside's p^aq^b theorem.

The theoretical contents will be presented in master classes following basic references in the bibliography. These lectures will be complemented with problem classes (classroom practice), in which students will apply the knowledge acquired in the theoretical lectures in order to solve problems. In the seminar sessions, exercises and representative examples will be considered. These will have been given to the students in advance, for them to have enough time to work out the solutions. Students must participate actively in the seminar sessions, and discussion of the solutions will be encouraged.

In some other classes the students will present work done in groups.

STUDENTS FOLLOWING CONTINUOUS EVALUATION



The final mark will be the weighted mean of the marks obtained in the following tasks:



T1. Individual problems or assignments along the course (with exposition in the classroom): 15%.

Some of these tasks will be presented in the problem sessions and some other in the seminar sessions. Attendance to seminar sessions is compulsory, except for good reason that will need to be documented.



T2. Problems or assignments done in groups along the course (with exposition in the classroom or in one of the teachers' office): 15%.



T3. Midterm exam (approximately in week 7 or 8 of the semester) of all the contents covered so far: 20%.



T4. Ordinary exam: 50%. There will be a written problem exam and a theory test that can be either oral or written. A minimum mark of 4,5 points out of 10 is needed in the ordinary exam in order to pass the course.



STUDENTS NOT FOLLOWING CONTINUOUS EVALUATION



In this case, 100% of the mark will correspond to the written ordinary exam. As a consequence, a minimum mark of 5 is needed in this exam in order to pass the course.

Basic bibliography

B. HUPPERT, Endliche gruppen I. Springer-Verlag, Berlín, 1967.

B. HUPPERT, Character Theory of Finite Groups. Walter de Gryter, Berlín, New York, 1998.

I.M. ISAACS, Character Theory of Finite Groups. Dover Publications, New York, 1994.

I.M. ISAACS, Finite Group Theory. American Mathematical Society, Providence (Rhode Island), 2008.

W. LEDERMANN, Introduction to Group Characters. Cambridge University Press, 2nd ed., Cambridge, 1987.

G. NAVARRO, Un curso de álgebra, Universidad de Valencia, 2002.

J. ROSE, A Course on Group Theory. Dover Publications, New York, 1994.

In-depth bibliography

J.L. ALPERIN, R.B. BELL, Groups and Representations. Springer, Berlin-New York, 1995.
L. DORNHOFF, Group Representation Theory, Part A. Marcel Dekker, New York, 1971.
L.C. GROVE, Groups and Characters. John Wiley & Sons, Inc., New York, 1997.
D.J.S. ROBINSON, A Course in the Theory of Groups, 2nd ed. Springer, New York, 1996.