## Cosets and Lagrange’s Theorem

Over the last couple of weeks I’ve been making my way cautiously through Unit GTA4 – I say cautiously because there are a couple of warnings in the text about how relatively difficult some students find the last section of this unit, so I wanted to make especially sure that I had a good understanding working with cosets before I tackled the intimidating Section 6.

Actually, I think I struggled more with Sections 2 and 3 than with the final section; there were a few questions that I just didn’t even know how to begin to answer (thankfully the solutions given at the end of the book did make sense when I eventually read them – so at least it wasn’t completely incomprehensible!).

Section 6 itself, which is about infinite quotient groups, was much more manageable than I expected – quite good fun, in fact! Earlier on in the unit, I was wowed by the fact that you can divide an infinite group, such as $(\mathbb{Z},+)$, into a finite number of cosets – I was even more impressed with the bit in Section 6 about dividing $\mathbb{R} / \mathbb{Z}$ into cosets that form an infinite quotient group. I’m not sure why exactly, but an infinite group having an underlying structure which is isomorphic to another infinite group pretty much blows my mind!

The question in TMA02 that relates to this unit was more straightforward than I expected; I think the questions based on GTA2 and GTA3 are a bit more challenging, although it could just be the fact that I’ve had a long time to get used to working with groups and subgroups now. So the first draft of TMA02 is now pretty much finished, bar any changes due to the feedback I’ll get from TMA01.

Now it’s on to the first unit of the Linear Algebra block, which is about vectors and conics. I must admit, I’m not particularly looking forward to this unit – I’d much rather get straight on with Group Theory Block B, but unfortunately I’ve got five units of linear algebra and four units of Analysis Block A to get through first!

## Permutations

I’m not a great fan of bank holidays, because I generally don’t like the disruption of my normal routines, but one thing bank holidays are great for is getting ahead with my study schedule! This week I’ve had loads of time to wrap up TMA01, and to work through Unit GTA3, which is about permutations.

I’d hoped that there would be some crossover between the combinatorics stuff that we covered in MS221 Unit B1, and the group theory permutations material in GTA3. It turned out that although there weren’t any explicit connections made between the two topics, an understanding of the basics of permutations (in the sense of questions like “how many different ways can I arrange the numbers 1, 2, 3 and 4?”) was definitely useful to have.

## Getting my teeth into TMA01

This week I’ve been making a start on TMA01, and so far I’m really enjoying it. The TMA is split into two parts, with two questions in Part 1 and four question in Part 2. Unfortunately for me, both questions in Part 1 involve lots of graph and diagram sketching! I found it a bit hard to get back into the swing of doing that kind of question, but I think I’ve just about got the hang of it now.

I’ve been dipping into Part 2 of the TMA too, while avoiding the bits that involve sketching diagrams (I thought I’d leave those bits til last, and get them over with in one big sketching extravaganza), and I particularly enjoyed the proof by contraposition question. I don’t remember covering this kind of proof in MS221, so maybe that’s why it’s entertaining me so much, but either way there’s just something very satisfying about proving $A \Rightarrow B$ by proving $\neg B \Rightarrow \neg A$.

Still, as much as I’m enjoying the TMA, I really can’t wait to get back to the GTA course units. I’m just at the start of Section 4 of GTA2 at the moment, and I’m loving it! I particularly liked the section about cyclic groups, and the video programme about isomorphisms (the bit where the presenter says “But first, let’s see what’s going on over here…” and then the camera pans across to a group of dancers doing some kind of Scottish formal dancing was brilliantly surreal. Even better than the “Symmetry counts” programme’s kitchen full of platonic solids!)

I’m very eager to get started on GTA3 too, because the unit is called “Permutations”, which gives me hope that it’s going to be something to do with combinatorics. Some of the exercises using two-line notation for symmetries reminded me a bit of finding all the possible combinations/permutations of the vertices in the figure, so I’ll be very happy if it turns out that you really can leverage combinatorics techniques to explore symmetries. At the very least, I hope the accompanying video programme is as bizarre and entertaining as the GTA2 programme!

## M208 course materials are here!

I wasn’t expecting these until early January, so I was pleasantly surprised when my M208 course materials arrived yesterday. The course books themselves are lovely, very nicely designed. I love the hexagonal tiling pattern on the covers, I’ve had a bit of thing for hexagons, ever since reading Finding Moonshine. This honeycomb pattern particularly appeals to me because it reminds me of three awesome things: bees, circle packing and Blockbusters.

And the contents of the books are even better! I’m so happy that I can finally get on with the Group Theory A units, I’ve been almost literally drooling with anticipation. I’ve started unit GTA2 (Groups and subgroups) today, and so far it’s been lots of fun. It did take me a little while to get used to the format of the example solutions, which have the proof of a result down the right-hand side and explanatory comments down the left-hand side, though. I wonder if those “training wheel” explanations will be omitted from the later units?

The group theory units come with some DIY papercraft models of the platonic solids, which gave me a chuckle since I’ve only just finished making my own models:

I might make the blue OU versions as well, you can never have too many platonic solids!

I’m really pleased to have my hands on the first assignment booklet too – it’s very nice to know exactly when the first four TMAs will be due, and exactly what each question covers. The first TMA is split into two parts: Part 1 covers Unit I1, and is due in on the 12th February, while Part 2 covers Units I2 and I3, and has a cut-off date of 5th March. The next few TMAs seem spaced out so that there’s one every four or five weeks: TMA02 is due in on 09/03/10, TMA03 on 21/05/10, and TMA04 on 25/06/10. It’s a bit intimidating, but hopefully it won’t be much more strenuous a schedule than the one I had to work to last year, for M366 and MS221. At least I’ll only have one exam to worry about!

## M366 exam results released!

I had to read and re-read this a few times to make sure I wasn’t hallucinating, but it turns out that I’ve somehow managed 80% on the M366 exam! And even more luckily, it seems that the examining board were in a particularly lenient mood, since they’ve decided that my overall course result is a Distinction, rather than the Pass 2 that it should technically have been. I’m overjoyed – I really thought I’d either fail or scrape a Pass 4 for this course. It’s a huge relief to know that I won’t have to resit the exam, or substitute a different Level 3 Computing course into my diploma and degree. Phew!

## Finding Moonshine

I registered for M208 this week, and I’m really looking forward to the delivery of my course materials next month, so I can carry on with the Group Theory A block. In the meantime, I’ve been reading another Marcus du Sautoy book, Finding Moonshine, which is a very apt choice to pass the time with, since it’s about group theory and symmetry.

Finding Moonshine is split into twelve chapters, one for each month of one year of du Sautoy’s life, and each chapter combines the history of group theory with more personal anecdotes about the author’s own career. I was a bit ambivalent about this approach at first, but over the course of the book I really warmed to it. I enjoyed getting a glimpse of what it’s like to be a professional mathematician; particularly the little details of du Satuoy’s working style, like the yellow legal pads he prefers to write on, and stories about the overseas trips he makes over the course of the year.

Actually, aside from the wonders of group theory, the main thing I got out of reading Finding Moonshine was an increased urge to travel! I particularly enjoyed the passages about the Alhambra, and I’m now determined to take a trip there myself, most likely in 2012 when I’ll hopefully be taking M336: Groups and geometry. I also really want to visit the glass pyramid at the Louvre, too. I can only imagine how awe-inspiring it must be to see it in person.

As for the mathematical content of the book, I think it’s more enjoyable in that respect than The Music of the Primes; I’ve certainly come away from Finding Moonshine with group theory fever, and I can’t wait to get back to studying it formally. I do think that the book might have been a bit over my head if I hadn’t already met the concept of symmetries as transformations in MS221, though, so perhaps this wouldn’t be a great choice for someone completely new to the subject. The next book on my reading list is Ian Stewart’s Why Beauty Is Truth: The History of Symmetry, so it’ll be interesting to see if it turns out to be a gentler introduction to the awesome world of group theory.

## MS221 exam results released!

Surprisingly, I managed to scrape a distinction with 87% in the exam, along with an Overall Continuous Assessment Score of 94% – I’m so relieved that I won’t have to resit this exam, and to be honest I was expecting a Pass 3 or maybe a Pass 4. Well done to everyone who sat the exam, and I hope you all got the grades you wanted!

## The Music of the Primes

I must admit, I haven’t done very much studying at all over the last couple of weeks. I can’t seem to gather up the motivation to finish Unit AA1, knowing that I’ll probably have to go through the entire unit again next year once the Analysis Block A assignment(s) is/are actually due. Instead I’ve been reading Marcus du Sautoy’s The Music of the Primes, and falling in love with prime numbers!

A couple of years ago I attempted to read a different book about the Riemann Hypothesis – I think it was Prime Obsession – and I just couldn’t get into it, so I was half-expecting to find myself bored by The Music of the Primes. Thankfully it was much a more compelling read than I anticipated! Or perhaps I’m just better equipped to understand the appeal of the primes these days.

The Music of the Primes is a really enjoyable book, and the only criticism I’ve got is that some of the material was quite familiar to me – the bit about cicadas and their prime-numbered breeding cycles, for instance – so perhaps if you’re a seasoned reader of popular maths books then you might find The Music of the Primes a bit boring. But otherwise, I’d recommend it to anyone interested in primes and their mysteries.

For me, the main attraction of prime numbers is that they are fundamental, and at the same time infinite; it seems odd that there are an infinite number of the building blocks out of which the integers are made. Imagine if there were an infinite number of chemical elements! One of my favourite daydreams is imagining the primes stretching out across the far reaches of the real number line, getting bigger and bigger, but more and more sparsely dotted around the line – on and on forever!

The other thing I like about primes is the fact that they’re so important in cryptography – it amuses me that the study of prime numbers has such big practical applications, and I suppose it goes to show that seemingly abstract and academic topics can yield unexpected concrete benefits. Although of course, not everything humans have used prime numbers for could be described as beneficial.

I wonder what the practical applications of group theory are? The next book on my reading list is Finding Moonshine, also by Marcus du Sautoy, so hopefully I’ll find out soon enough!

## AA1 and thinking like a mathematician

This week I’ve been dabbling a bit in one of the other M208 units available on OpenLearn, AA1: Numbers. I think this unit is supposed to be studied after all the Group Theory A and Linear Algebra books, but it seems pretty straightforward so hopefully studying it out of sequence won’t do me too much harm. In a way, the material in AA1 is quite familiar but at the same time it seems like we’re looking at these subjects in a more precise, rigorous way than in MS221. It’s nice, but a bit intimidating! I often get a bit anxious that what seems like a simple statement might have some deeper meaning which is sailing over my head. Still, if that’s the case I’m sure I’ll find out pretty quickly when it comes to the assignments.

## Getting back into group theory

I was kind of expecting M208 to be over my head, but so far I’m really surprised at how much material in the first few units of M208 is also covered in MS221, and at how gentle an introduction the Intro Block actually is. Very pleasantly surprised indeed! I’ve been working on Unit I3: Number systems and GTA1: Symmetry this week, and there were only really two new ideas introduced in these, so I feel like I’m getting a nice, steady warm-up session before the really hard stuff starts.

The topic in I3 that I’d never encountered before was equivalence relations. It took me a little while to get my head around the idea (and to stop thinking of the phrase “equivalence class” in its software testing context from M255), but I think I’ve got a fairly good grasp of it now. I even managed to give a coherent explanation of it to Alex, with the help of the “coloured blocks” example from the h2g2 Equivalence Relations page.

The unit about Symmetry introduces the two-line symbol notation for describing symmetries, which I’d never come across before, and to be honest at first I couldn’t really see the point of doing it this way. If I’m going to describe a reflection of a square in the vertical axis, for instance, I’d much rather write ${q_{\pi /2}}$ than

$\left( {\begin{array}{*{20}{c}}1 & 2 & 3 & 4 \\4 & 3 & 2 & 1 \\\end{array}} \right)$.

But later on in the unit, I watched the video section about symmetries of Platonic solids, and it certainly sounds a lot easier to write down the symmetries of a tetrahedron in a two-line symbol than trying to figure out how to represent it in the form ${q_\theta }$!

Speaking of Platonic solids, I absolutely love the animations of them on the Wikipedia Platonic solid page. I could watch this octahedron spinning all day!

Octahedron image by Cyp, under the Creative Commons Attribution ShareAlike 3.0 License