Algebra

Code:

M2032

Acronym:

M2032

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	9	72	243
L:CC	0	study plan from 2021/22	2	-	9	72	243
			3
L:F	0	Official Study Plan	2	-	9	72	243
			3
L:G	0	study plan from 2017/18	2	-	9	72	243
			3
L:M	0	Official Study Plan	2	-	9	72	243
L:Q	0	study plan from 2016/17	3	-	9	72	243

Teaching language

Portuguese

Objectives

To get acquainted with basic concepts of group theory and ring theory. To understand in particular the importance of group actions and the Sylow theorems.

Learning outcomes and competences

Capability of solving problems in the area. Autonomy on solving exercises.

Working method

Presencial

Program

1. PERMUTATIONS AND GROUPS. Permutations, even and odd. Groups, symmetric groups, alternating groups and symmetry groups.

2. SUBGROUPS AND HOMOMORPHISMS. Subgroups, Lagrange's Theorem, quotient groups, normal subgroups, homomorphisms, isomorphism theorem, Cayley's Theorem, cyclic groups, direct product, conjugation.

3. GROUP ACTIONS. Definition and examples, orbits and stabilizers, orbit-stabilizer formula, class equation, applications.

4. SYLOW'S THEOREMS. Sylow's Theorems and their applications.

5. FINITE ABELIAN GROUPS. Fundamental theorem of finite abelian groups. Reference to the finitely generated case.

6. RINGS AND FIELDS. Rings, unitary and commutative rings, fields, subrings, ideals, quotient rings, homomorphisms, Isomorphism theorem.

7.INTEGRAL DOMAINS. Integral domains, characteristic, divisibility, euclidean domains, principal ideal domains, unique factorization domains.

Mandatory literature

John B. Fraleigh; A first course in abstract algebra. ISBN: 978-1-292-02496-7

Complementary Bibliography

Joseph J. Rotman; An^introduction to the theory of groups. ISBN: 0-387-94285-8
Rui Loja Fernandes; Introdução à álgebra. ISBN: 972-8469-27-6

Teaching methods and learning activities

Teaching methods and learning activities

The contact hours are distributed in theoretical and theoretical-practical classes. In the first ones, the contents of the program are studied, often using examples to illustrate the concepts treated and to guide the students in the resolution of exercises and problems. In the theoretical-practical classes, exercises and problems are solved, which are indicated in advance for each week. List of exercises and other course materials are available on the course page at Sigarra.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

No requisites.

Calculation formula of final grade

The syllabus will be divided into two parts. each one evaluated by a test worth 10 points.

The second test is held simultaneously with the first season exam. In the same occasion, it is possible to repeat the first test, and the marks obrtained prevail for those students wishing it.

First season exam:

1. The final mark is the sum of the marks obtained in each test, except possibly in the following case:

2. Marks above 18 require an extra proof (oral or written).

Second season exam:

1. In the second season exam, students may repeat both tests or just one of them (except when they are just trying to improve their mark).

2.The mark of one (but only one) of the tests may be replaced a posteriori by the mark obtained in the respective test of the first season, in the version which favours most the student (except when they were approved before).

3. The final mark of the second season is the sum of the marks obtained in both tests, rounded to integers, except possibly in the following cases:

4. Students having obtained a mark equal or above 8,0 and below 9,5 have access to a complementary proof to decide if they are approved (with 10 points) or if they fail (with 8 or 9 points).

5. Marks above 18 require an extra proof (oral or written).

Special assessment (TE, DA, ...)

Examinations required under special statutes shall consist of a written test that may be preceded by an oral test, to assess if the student satisfies minimum conditions to attempt to obtain approval at the discipline in the written test.

Classification improvement

In an examen meant to improve a positive marking, it is not possible to use partial markings previously obtained.