Essay instructions
The essay is supposed to be a relatively short (about 2–5 pages),
clear description about a given topic. Extensive use of mathematical
symbols and formulae is discouraged. Proofs are not required; it is more
suitable to describe the ideas of the theorems involved and give some
examples. Also the definitions should be explained informally, as if you
were teaching new algebraic notions to someone who already knows the
basics.
Anything at hand can be used as a reference, including the course notes.
Direct copying is of course forbidden. (I will check the Wikipedia.)
Topics
The essay can be written during either the first or the second half of the
course. There are different topics for each period. The deadline on the
first half is one week after the end of the week between periods, that is,
19th March (Minna Canth celebration day). The deadline on the second half is the last lecturing day, that is,
7th May.
List of topics for the first half (deadline 19/3):
- Decomposition of mappings
- Dihedral groups
- Small symmetric and alternating groups
- Symmetry groups in the Euclidean space (*)
- Group products
- Finite simple groups
- Composition factors of groups
- Sylow's theorems
- Colouring problems (*)
- The life and legacy of William Burnside (*)
List of topics for the second half (deadline 7/5):
- Division rings
- The tensor product
- Universal properties (*)
- The quaternions (or octonions)
- Lie algebras
- Finite fields
- Algebraic extensions
- Geometric constructions
- Solving polynomial equations (*)
- The life and legacy of Évariste Galois
You can choose any topic you like. Star (*) in the end of a topic means
that the lecturer has additional material covering that topic.
Assessment
The essay is marked on the scale 0–3. Zero marks means failure. The
requirements for each mark are approximately the following:
- 1 mark: Mainly grammatically correct writing, related to the given
topic.
- 2 marks: Clear and intelligible composition about the given topic.
Illuminating descriptions and examples.
- 3 marks: Instructive and coherent exposition that has numerous
illuminating examples. The language is rich and variable.
Note that a lengthy text does not directly effect a good mark.
The essays will be returned to the students later, with comments.