Course content

The course deals with groups and rings. A group is one of the simplest mathematical objects one studies in modern mathematics, and they appear in a great many contexts. They are building blocks for symmetries, patterns and transformations. Whether you study molecules, crystals, or Rubik's cube, the underlying mathematical structure is a group.

The course gives examples of the axiomatic framework of mathematics, and it deals with groups, subgroups, homomorphisms of groups, cyclic groups, permuation groups, finite abelian groups, the fundamental theorem for finite abelian groups, normal subgroups, factor groups, group action on a set, Sylow theorems, elementary ring and finite fields, vector spaces over an arbitrary field, combinatorial counting results, elementary number theory with applications to linear congruences and codes.

Learning outcome

1. Knowledge. The student is familiar with fundamental concepts and methods of group theory, including normal subgroups and factor groups, permutation groups and cyclic groups. The student has knowledge about the classification of cyclic and abelian groups, about Lagrange's and Cayley's theorem, in addition to Burnside's theorem for group actions. The student knows the fundamental theorem for group homomorphisms and the Sylow theorems. The student also knows the fundamental concepts from elementary ring theory with emphasis on commutative ring theory, and field theory and vector spaces over arbitrary fields. The student has some knowledge of application of group theory in enumeration and number theory in secret codes (RSA). The student has some knowledge about the axiomatic framework within mathematics, in addition to basic logic concepts and different structure of proofs.

2. Skills. The student is able to recognize group and ring structures and group actions, and to analyze simple aspects of these. Furthermore the student masters the theoretical base of application of group theory in enumeration and the number theoretical base for secret codes (RSA). In addition the student can preform elementary mathematical proofs.

Learning methods and activities

Following teaching methods and learning activities are possibly going to be use in this course: Flipped classroom teaching, ordinary lectures, group activities and exercises. The course work will contain activities where the students train on (1) absorb and understand mathematical knowledge on their own, and (2) to communicate mathematical knowledge and understanding to others.

Point (1) could be implemented by the students first trying to prove/understand a result/concept on their own, and then being presented/discussed in groups or in plenum. Point (2) could be implemented through work in small groups, presentation of solutions of problems, of proofs of mathematical results og of new mathematical concepts.

Recommended previous knowledge

MA1301 Number theory, MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications, TMA4110/4115 Calculus 3 or equivalent.

Course materials

Will be announced at the start of the course.

Credit reductions