Groups and Symmetries

Recommended Prerequisites

Number Theory, Linear Algebra and Analytic Geometry.

Teaching Methods

In TP classes, the fundamental  results of elementary group theory are presented with rigour and detail, being followed by examples of application (theoretical and practical) to test whether they have been understood. In PL classes and as homework, students must solve many proposed exercises with various degrees of difficulty. Independent solution of problems is greatly encouraged.

During the semester, students may use tutorial time to clarify their difficulties and to help them training their skills in problem solving.

Learning Outcomes

The acquisition of theoretical and practical  knowledge of elementary group theory. To know how to put in good use  fundamental results  in order to produce proofs and to solve problems.

The main competencies to be developed are: ability for  generalization and abstraction; ability to formulate and solve problems; ability to look for the best way to use the known results; critical thinking.

Work Placement(s)

No

Syllabus

Groups. Subgroups and generators. Permutations. Homomorphims, Isomorphisms. Plato’s Solids and Cayley’s Theorem. Matrix Groups. Products. Lagrange Theorem. Cauchy Theorem. Conjugacy. Quotient groups. Actions, orbits and stabilizers. Finitely generated abelian groups. The Sylow Theorems.

Head Lecturer(s)

Cristina Helena de Matos Caldeira

Assessment Methods

Assessment
Exam (100%) or Midterm exam(75%) + Problem resolving report (25%): 100.0%

Bibliography

M. A. Armstrong, Groups and Symmetry, Springer Undergraduate  Tests in Mathematics (1998)

 

M. Sobral, Lecture Notes (Webpage of the course)