Academic Year 2023 - 2024
Semester 2
This course introduces basic concepts in group theory. Major topics: Modular arithmetics. Binary operations. Groups. Subgroups. Group homomorphisms. Examples of groups Symmetric groups and Cayley’s theorem. Cyclic groups. Cosets and Theorem of Lagrange. Fermat’s Little Theorem and Euler’s phi function. Direct products of groups. Normal subgroups. Quotient groups. Isomorphism Theorems. Group actions. Stabilisers and orbits. Examples and applications. Major topics: Divisibility, congruences. Permutations. Binary operations. Groups. Examples of groups including finite abelian groups from the study of integers and finite non-abelian groups constructed from permutations. Subgroups. Cyclic groups. Cosets. Theorem of Lagrange. Fermat’s Little Theorem and Euler’s Theorem. Direct products of groups. Normal subgroups. Quotient groups. Isomorphism Theorems
I took this class because I was unsure if I wanted to go down the Algebra route for my math degree. This class is taught by Prof Chin Chee Whye (CCW) and he is infamous for setting extremely hard papers. I think this iteration was not an exception.
This is essentially an introductory group theory course but I think there was some additions as well. The previous iterations under another Prof did not have group actions but we do under CCW. Essentially if you run out of tools in your basic group theory, you then rely on group actions, cook up some homomorphisms and use certain properties to your advantage to prove something that seem so far fetched.
Homework took up very long for me. That is because I was very disciplined and REFUSED to check solutions or find something similar online. Hence, my paper usage for the 5 homeworks (ok 4, the first homework is easy), is ridiculous. Hard work paid off when I scored full marks for all my homework. Its a great sense of ahievement.
One term that keeps getting thrown around was universal property. I can see why it is useful, but I would want to get some examples on how it can be useful. Undergraduate group theory textbooks won’t feature universal property. These pop up usually in books talking about Category Theory. Prof CCW sure loves his “god given homomorphisms”.
For the exams, the midterms was not that hard. First 2 questions were routine, the last 3 is really just examples and techniques. You know, you know. You don’t, then tough luck. I think if group theory ain’t intuitive for you, then the struggle is real. For finals, there were 10 homework-style questions which had to be completed in 2 hours without any cheat sheet. This can be good and bad because it forces people to just internalise everything, but if you are stressed, then your mind can go blank. Personally, I think I did 5 questions properlly. The rest of the other 5 had little to no progress. I think given how much time I spent on the homework, completing the final is very tough. For PYPs, it is very hard to say because they were considerably easier and I could even do all the S-level papers. When I saw the S-level finals (MA2202S , the 5 MC version of Algebra I), I felt that I could do the first 4 questions within 30 minutes. I thus personally think this class under CCW should have been 5MC instead of 4MCs. Again, no expectations. I think I am very fortunate (yes I know, ironic) to have CCW has a prof for MA2202 because I learnt so much cool things the other cohorts missed out on. Though, I question if it was really necessary to put 10 questions into the final. Finishing it was practically impossible for a person who just learnt group theory for the past 3 months.
Here are some quotes of CCW which can be found on other reviews.