SPECIFIC COMPETENCES



M01CM01 Understand what an abstract group is from known examples of groups in other courses: groups of numbers, residue classes, matrices, etc.



M01CM02 Know the basic concepts in group theory (subgroups, normal subgroups, factor groups, homomorphisms,...).



M01CM03 Know how to operate with elements in some important groups (cyclic groups, direct products, permutation groups,...) and their main properties.



M01CM04 Understand the basic concepts in the theory of rings and fields (subrings, ideals, quotients, homomorphisms, field characteristic, field of fractions,...).



M01CM05 Understand the properties of divisibility of univariante polynomials and, in particular, the use of the main irreducibility criteria.



LEARNING RESULTS



To know the basic concepts of group theory (subgroups, normal subgroups, quotients, homomorphisms ...) and the properties of the most important groups (cyclic, direct products, dihedral, symmetric, ...).



To know the basic concepts of the theory of rings and fields and, in particular, of the rings of polynomials in one and several indeterminates.

1. GROUPS. FUNDAMENTALS: Concept of group. Examples (groups of numbers, Z/nZ and its units, groups of matrices, groups of symmetries,...). Subgroups. Subgroup generated by a set. Cosets and index of a subgroup. Lagrange's Theorem. Products of subgroups. The order of an element. Cyclic groups.

2. NORMAL SUBGROUPS AND GROUP QUOTIENTS: Conjugacy and its properties. Normal subgroups. Construction of group quotients. Subgroups of a group quotient.

3. GROUP HOMOMORPHISMS: Group homomorphisms. The kernel and the image of a group homomorphism. Isomorphic groups. The Isomorphism Theorems.

4. CYCLIC AND ABELIAN GROUPS: The subgroups of a cyclic group. Direct products. Classification of the abelian finite groups. Classification of some groups of small order.

5. THE SYMMETRIC GROUP: Permutations, decomposition in disjoint cycles. Signature. The symmetric and alternating groups. Conjugacy in the symmetric group. Cayley's Theorem. Simplicity of the alternating groups.

6. RINGS AND FIELDS: Rings and fields, first properties. Characteristic and prime field. Integral domains. The field of fractions of an integral domain . Subrings, ideals and ring homomorphisms. Maximal ideals and fields. The Chinese Remainder Theorem.

7. UNIVARIATE POLYNOMIALS: Factorization of univariate polynomials. Irreducibility criteria. Quotients of polynomial rings. Finite fields.

Masterclasses, seminars and problem sesions. Students must participate actively in class solving the proposed problems.

There will be two written exams: one partial and one final. The final mark will take into account tha student's attitude in his/her learning process. It will be calculated averaging the marks in the different activities according to the following weights:

- 60-80% final written exam.

- 10% partial written exam.

- 10-30% classroom work and individual or group homework.

To pass the course a mark of at least 4 points out of 10 in the final exam is required.

None.

Basic bibliography

J.D. DIXON, Problems in Group Theory. Dover, 1973.

S. LANG, Undergraduate Algebra, 2nd ed. Springer, New York, 2001.

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

A. VERA; F. VERA, Introducción al Algebra, I. Ellacuría, Bilbao, 1984.

A. VERA; F. VERA, Aljebrarako Sarrera, I. Ellacuría, 1991.

A. VERA; J. VERA, Problemas de Algebra, I: Teorías de Grupos y de Cuerpos. AVL, 1995.

In-depth bibliography

J. F. HUMPHREYS, A Course in Group Theory. Oxford University Press, 1996.
I. M. ISAACS, Algebra. A Graduate Course. Brooks/Cole Publishing Company, Pacific Grove, California, 1994.
H. KURZWEIL; B. Stellmacher, The Theory of Finite Groups. An Introduction. Universitext, Springer, New York, 2004.
J.S. ROSE, A course on Group Theory. Cambridge University Press, 1978.

Journals

This is an introductory course, so no periodic publication is recommended.