Apr '26 Notes
This is my fourth set of monthly notes for this year where I write down interesting facts and ideas I have explored during my spare time. There were three things in particular that occupied my leisure time this month. First, I managed to learn the proof of Tutte's famous theorem that any \( s \)-arc-transitive finite cubic graph must satisfy \( s \le 5. \) I learnt the proof from Norman Biggs's book Algebraic Graph Theory. The original proof appears in Tutte's 1947 paper 'A family of cubical graphs' (DOI). Biggs's presentation differs considerably from Tutte's original argument and relies heavily on the properties of stabiliser sequences of arcs. I should say that Biggs's proof, while complete, is extremely condensed. The proof reads more like a high-level outline that moves rapidly from one main result to the next without sufficiently explaining the intermediate steps. As a result, it took considerable effort to work out the proofs of the intermediate results. Biggs presents the proof in roughly nine pages spread across two chapters. However, when I worked through it in full detail while ensuring that every step is justified, my notes eventually grew to around 18 pages of A4 paper. The proof is quite involved, so I have not included it in these notes. Perhaps someday, when I have more time, I will distil my handwritten notes and publish them here on my website.
That was the first thing I spent time on this month. The second was revisiting some elementary results in group theory concerning cosets. I have found cosets to be an extremely useful concept that plays a central role in many areas of mathematics, including coding theory, Galois theory, field extensions and graph theory. In fact, Biggs's proof of Tutte's theorem discussed above also relies substantially on the theory of cosets. Since these results are relatively elementary and easier to write up, they are included in this set of notes.
Apart from mathematics, I also spent part of my spare time improving my new web project named Wander Console. It is a tiny, decentralised, self-hosted web console that allows visitors to a website to explore other interesting personal websites. It is similar to the now-defunct service named StumbleUpon, but unlike StumbleUpon it has no central service and no server-side logic. Wander is hosted entirely on independent personal websites. Wander Consoles link to one another and fetch web page recommendations from each other. The entire tool consists of just two files: an HTML file and a JS file. Everything, including connecting to other Wander Consoles in the network and recommending webpages, happens entirely on the client side in the user's web browser. See the project README for more details.
Contents
Coset Results
This section presents some very elementary results about cosets, together with brief proofs. These results appear repeatedly across many areas of mathematics and I often find myself using them almost instinctively, without consciously thinking through the underlying arguments each time. But once in a while, I like to sit back and ponder about their proofs from first principles, reflect on why they work and appreciate their elegance. The subsections below collect some of those proofs.