MM515: Groups and Vector Spaces (5 ECTS)

STADS: 13001501

Level
Bachelor course

Teaching period
The course is offered in the spring semester.
4th quarter.

Teacher responsible
Email: hjm@imada.sdu.dk

Timetable
Show entire timetable
Show personal time table for this course.

Comment:
Ubegrænset deltagerantal. 4. kvartal.

Prerequisites:
None

Academic preconditions:
Introductory linear algebra and a basic knowledge of rings and elementary number theory.

Course introduction
The course introduces basic topics in group theory and extends previous insights into vector spaces and fields.

Qualifications
Having completed the course successfully the student will have acquired
- insight into the mathematical treatment of symmetries
- extended knowledge of abstract axiomatic reasoning and its importance in linear algebra
- experience of the interplay between different mathematical structures in the description of a mathematical problem

Expected learning outcome
After the course the student is expected to

• (after 1/2 hour's preparation time) select essential definitions and results from a given subtopic of the relevant course material and present these in a precise mathematical language including formal proof(s) of at least 1 essential (sub)result. This part of the examination is expected to last 15 -10 minutes.
• (without time for preparation) give correct and exhaustive answers to questions concerning central definitions and/or results chosen among all topics treated in the course. This part of the examination is expected to last 5 - 10 minutes and proofs can be touched upon only in a very sketchy way.

Subject overview
- Groups:
Definition, basic examples and properties, subgroups, isomorphism, homomorphism, congruence, Lagrange's Theorem, normal subgrup and quotient group, the Homomorphism Theorems, symmetric and alternating groups, direct products and finite abelian groups, group actions and the Sylow Theorems.

- Vector Spaces:
Sum, direct sum, and quotient spaces. The Homomorphism Theorems.

- Interplay between groups and vector spaces:
Vector spaces over finite fields.
Linear groups.
Linear representations of groups.
Field extensions as vector spaces. Galois Theory.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
Oral exam. Eksternal examiner. Marks according to the Danish 7-scale.

Reexamination after 2. quarter.

Expected working hours
The teaching method is based on three phase model.

Forelæsninger (28 timer) og øvelser (18 timer).
Educational activities

Language
This course is taught in English, if international students participate. Otherwise the course is taught in Danish.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.