Group Theory

Code:

M2027

Acronym:

M2027

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

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

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	-	6	56	162
L:CC	9	study plan from 2021/22	2	-	6	56	162
			3
L:F	2	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

To introduce the concepts, methods and basic results of Group Theory.

Learning outcomes and competences

Upon completing this curricular unit, the student should:

(1) master the basic concepts, methods, and results of Group Theory;

(2) to be able to analyze and solve problems within Group Theory, using the methods and results that best apply to the problems under study;

(3) to appreciate the connections of Group Theory with other areas of mathematics such as Geometry;

(4) be able to efficiently and clearly communicate their resolutions of problems, and their understanding of the subject.

Working method

Presencial

Program

• Basic notions: binary, order and equivalence relations.


• Definitions and elementary properties of groupss; important examples of groups: permutation groups; the integers, the real and the complex numbers; integers modulo n; permutations; matrices and linear groups; symmetry groups.


• Direct product of groups.


• Subgroups.


• Generators of a group; cyclic groups; cosets; Lagrange's theorem and some of its consequences, such as Fermat's “little” theorem and its generalization by Euler, as well as the fact that every prime-order group is cyclic.


• Homomorphisms and isomorphisms. Conjugation. Cayley's Theorem.


• Fundamental theorem of finitely generated abelian groups.


• Normal subgroups and quotient groups. Fundamental theorem on homomorphism for groups.


• Group actions and applications. Sylow's Theorems.

Mandatory literature

Gregory T. Lee; Abstract Algebra: an Introductory Course, Springer, 2018. ISBN: 978-3-319-77648-4

Complementary Bibliography

Fernandes Rui Loja; Introdução à álgebra. ISBN: 972-8469-27-6
Rotman Joseph; A first course in abstract algebra. ISBN: 0-13-011584-3
B. L. van der Waerden; A history of algebra from Al-Khwarizmi to Emmy Noether. ISBN: 3-540-13610-X
Fraleigh John B.; A first course in abstract algebra. ISBN: 0-201-16847-2
Peter M. Neumann; Groups and geometry. ISBN: 0-19-853451-5
M. A. Armstrong; Groups and symmetry. ISBN: 0-387-96675-7
J.S. Milne; Group Theory, 2017

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.

keywords

Physical sciences > Mathematics > Algebra > Group theory

Evaluation Type

Evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

There are no rules concerning attendance frequency.

Calculation formula of final grade

Approval in the course unit may be obtained through a test or  in the final exam.

The test, to take place on a date to be announced, will take two hours.

Students approved in the test are exempt from participating in the final exam but may still participate in it.  The final grade is the maximum between the grades obtained in the test and in the exam.

In the "appeal" period the grade is classification obtained in it.

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.

Observations

As classes are given simultaneously with those of Algebra (M2032), with 9 ECTS, they are concentrated in the first 8 weeks of effective teaching in the semester. Student assessment on group theory topics is the same in both courses.