Category Archives: Group Theory

The Last Group Theory Feedback

My fifth assignment has been returned! And with it, the usual amount of feedback!

In this assignment, I failed to realise that a subgroup is normal if it is a union of conjugacy classes. So rather than form subgroups and state they were a union of conjugacy classes, I went through the process of proving each group was a sub group.  Too long-winded, and frankly a waste of time. (for reference, this refers to Theorem 3.5 on properties of subgroups, p.74 in the handbook for M208).

There was another question where I had to show that a particular group had no subgroup of order 4. I took this rather literally, and found the only 4 possible combinations of elements that could be a subgroup of order 4, and then individually proved each one did not qualify either as a subgroup or as normal.  Again, although I received full marks for this as it was correct, it was long-winded. It seems that all I needed to notice that a group of order 4 is either a typical cyclic group of order 4 ( C_{4} ), or a Klein group ( K_{4} ). A property of C_{4} is that it contains an element of order 4. But the question says the group has no subgroup of order 4! Therefore, my possible subgroups must be isomorphic to K_{4}. Property of K_{4}: it contains three elements of order 2. This cuts the number of possibilities in half. Amazing how mindful you have to be of the properties of everything...

When having to write down subgroups of a matrix group I thought I'd written two different subgroups, but they were actually the same. One of the entries in one subgroup was p-\frac{1}{p}, and the other was \frac{1}{p}-p. Now, these ARE both different. But they don't constitute as different general entries in my matrix, because they are both in \mathbb{R}-\{0\}. The correct answer had the second matrix with an entry that read 2\left(p-\frac{1}{p}\right), which of course has a different domain: 2\mathbb{R}-\{0\}, so it forms a different sub group. My brain had clearly just seen two different formulas and had gone "there we go, they're different!". Not so. Domains for subgroups must be checked!

This is incredibly easy to forget: When trying to prove something is not true, don't leave it at the generalised proof that it's not true. If it's not true, provide a specific counter-example. With numbers! Remember those? The things that you count with that aren't letters? Duh! Yes, so I know to be vigilant now...

I had some trouble explaining a mapping of complex numbers in English. This is problematic. I could've programmed it and shown you the mapping and in an instant you would've thought "ah yeah!". But with lengthy explanations about how we get to the point where the real part gets mapped to the inverse of the imaginary etc etc... things can get a bit muddled. It seems I need to resort to explaining things mathematically more often, using the facts I already have.

I need to do more reading on the Isomorphism Theorem (specifically with regard to the domain of the Image that the quotient group is isomorphic to). Wow, that was a bit wordy... Ultimately I need to be more mindful about domains of Images...

So that's it! Again, happy with my high mark, but lots to remain mindful of...

Group Theory Returns (with the Triforce)

I've just finished the first draft of my second Group Theory assignment. I am exhausted. I think of all my assignments, this is my most logic-dense. Just proof-reading the damn thing is causing brain burn. Though doing this kind of thing when you're tired never works, so I'll look at it over lunch tomorrow.

Some of the more complex concepts are so abstract and don't follow intuitively. Some do, but others don't. It's been fun, but I'm glad I don't have to dive any deeper into Group Theory right now.

What made this weekend's work more difficult was the recent release of Nintendo's Zelda: Breath of the Wild. Trying to concentrate on my study knowing that game was underneath my tv was tricky. What was particularly unfair was turning the page in my coursework to see they'd set a question about the Triforce.

UNFAIR. THANKS A LOT MATHEMATICS.

(also, #maybeTooMuchZelda)

Group Theory Feedback

My second assignment has been marked! Very happy with these results. Though as usual, my tutor has been great and filled my paper with suggestions on how to improve further.

The identity axiom for a group: Often it's really obvious to see that an operation is commutative. Really easy. So easy in fact, that it's often just as easy not to mention that it's commutative. In doing so, you kind of miss out half the answer. Always check! If it works one way, always prove it works the other too!

Another obvious thing that's easy to miss out... mentioning that your result does actually lie within the required working set. ie: if you're working in the \mathbb{Z} universe, you need to explicitly say that your result is also in \mathbb{Z}.

All transformations are relative! I sped through these questions without thinking... silly really. I slipped up here, and never mentioned the point around which something was rotated, or the point around which a reflection line was rotated.

Students apparently screw this up a lot... me included it seems... but answers should be in their correct forms. I was so used to writing Cayley/Group tables as answers, I neglected to realise that the question actually wanted the set which formed the group. Here, effort was spent where it didn't need to be.

Lastly, I need to get better at quickly being able to spot if a Cayley table is Abelian (commutative). This was a silly oversight on my part. Something that's a little less obvious is how to quickly find a group that is isomorphic to my initial (Abelian, in this case) group. I suppose this will come with time and familiarity!

Next up we've got linear algebra. This looks like a big section, so it's good that I'll have the Christmas holidays to break the back of it!

Finally, More Group Theory

I've just come to the end of the second section of study: Group Theory! It was great to get back to this. Since my introduction to it a few modules ago I had been really curious as to what the next steps were with it all, and it's really fun!

Though now I'm back in the same situation as I was when I was introduced to it: I just want to learn more! Group theory will now be shelved until about March I think (like analysis, there are two group theory sections).

Something I noticed though, even group theory at this level is starting to reflect what Alcock was saying in her book about analysis and the building of theorems to prove another further theorem. As such, I may have benefitted from creating that spider diagram I was such a big fan of before. Next up is linear algebra, but after that is the first section of analysis. I'll have to go into that with the view to actually making that spider diagram to get an overview of how all the concepts build on each other.

Year Three

My third module, completed last June, took me by surprise. As I'd started my statistics In February 2014, and completing in September 2014, I was hoping for a nice break before starting this third module.

The module I needed to take was MS221, rather innocently called "Exploring Mathematics". However, it transpired that the class that would be starting in September 2014 was to be the last ever of MS221. Normally this wouldn't be a problem, but MS221 followed on directly from material presented in my first module, and I was told I had to take this course. If I waited to take the replacement course I would face a ton of catch-up material .

So in summary, due to the slight course re-structure, I ended up not stopping to have a break between my second module and my third. This meant I was studying non-stop from February 2014 to July 2015. Exhausting.

I did manage to take a long holiday at the beginning of 2015, but tried to make sure I was several weeks ahead of my study before doing so.

Although "Exploring Mathematics" sounds rather unassuming, it covered some rather in-depth topics. On the one hand I was presented with proof by mathematical induction (which only proved itself to be baffling) but on the other I was also presented with group theory which describes a really nice formal form of symmetries. Seriously cannot wait to learn more about group theory...

Once July arrived, and I had taken my final exam I was once again a free man. Finally free to enjoy a Summer.

So that brings us up to date! I'm now enjoying my first proper break away from mathematics in 17 months, and the materials for my next module are due to arrive in under a month!