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. These notes include some updates about this tool.

{{ tocn }}

## 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 these little proofs.

### Subgroup

Definition. Let \( G \) be a group with operation \( \cdot \). Then a subset \( H \subseteq G \) is called a subgroup of \( G \) if \( H \) itself forms a group under the same operation and we write \( H \le G \).

### Cosets

Definition. Let \( G \) be a group, \( H \le G \) and \( a \in G \). The left coset of \( H \) by \( a \) is the set \[ aH = \{ ah : h \in H \}. \] Similarly, the right coset of \( H \) by \( a \) is \[ Ha = \{ ha : h \in H \}. \]

### Coset Membership

Theorem. Let \( G \) be a group with identity \( e \), \( H \le G \) and \( a \in G \). Then \( aH = H \) if and only if \( a \in H \).

Proof. Suppose \( aH = H \). Then \[ aH = H \implies ae \in aH = H \implies a \in H. \] Conversely, suppose \( a \in H \). Let \( x \in aH \). Then \( x = ah \) for some \( h \in H \). Then by the closure property of the subgroup \( H \), we get \[ a, h \in H \implies ah \in H \implies x \in H. \] Thus \( aH \subseteq H \). To show the reverse inclusion, let \( x \in H \). Since \( a \in H \), we have \( a^{-1} \in H \), so \[ x \in H \implies a^{-1} x \in H \implies a(a^{-1} x) \in aH \implies x \in aH. \] Therefore \( H \subseteq aH \). As a result, \( H = aH \).

### Coset Equality

Theorem. Let \( G \) be a group with identity \( e \), \( H \le G \) and \( a, b \in G \). Then \( aH = bH \) if and only if \( a^{-1} b \in H \).

Proof. Suppose \( aH = bH \). Then \begin{align*} aH = bH & \implies b \in bH = aH \\ & \implies b = ah \tag{for some \( h \in H \)}\\ & \implies a^{-1} b = h \\ & \implies a^{-1} b \in H. \end{align*} Conversely, suppose \( a^{-1} b \in H \). Let \( h = a^{-1} b \). Then \begin{align*} h = a^{-1} b & \implies ah = b \\ & \implies (ah)H = bH \\ & \implies a(hH) = bH \\ & \implies aH = bH \tag{since \( hH = H \)}. \end{align*}

Corollary. Let \( G \) be a group with identity \( e \), \( H \le G \) and \( a, b \in G \). Then \( aH = bH \) if and only if \( a \in bH \).

Proof. \[ aH = bH \iff b^{-1} a \in H \iff b (b^{-1} a) \in bH \iff a \in bH. \]

### Partitioning a Group into Disjoint Cosets

Theorem. Let \( G \) be a group with identity \( e \), \( H \le G \) and \( a, b \in G \). Then either \( aH \cap bH = \varnothing \) or \( aH = bH \).

Proof. Suppose \( aH \cap bH \ne \varnothing \). Then there exists some element \( x \in aH \cap bH \). Then \[ x = ah_1 = bh_2 \] for some \( h_1, h_2 \in H \). Therefore \[ a^{-1} b = h_1 h_2^{-1} \in H. \] Then by section Coset Equality, \[ aH = bH. \]

## Computing

### Wander Console Updates

Wander Console is a new project I developed in March while taking a short break from my algebraic graph theory studies. 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 codeberg.org/susam/wander for more details.

Two releases of Wander were made this month. One notable feature introduced this month is the console network crawler. You can visit any Wander console and use this feature to crawl the network reachable from it. To see this in action, go to my console at wander/, click the Console button at the top and then click the Crawl button.

Since different consoles link to different sets of peers, each one has its own neighbourhood, so the crawler output varies from console to console.

Due to the decentralised nature of the tool, it is difficult to know exactly how many people have set up Wander Console on their websites. Nevertheless, I used the crawler with a known set of consoles to explore as much of the network as possible. The result is available at susam.codeberg.page/wcn/. It currently shows over 50 consoles recommending more than 1400 web pages from the small web of personal sites. For a project that is only six weeks old, these seem like decent numbers.

### Flush Output

When I log into my Debian 13.2 system via SSH, I find that `ctrl`+`o` does not enable output discarding.

    $ stty -a | grep 'discard\|flush'
    werase = ^W; lnext = ^V; discard = ^O; min = 1; time = 0;
    echoctl echoke -flusho -extproc
    $ ping 127.0.0.1
    PING 127.0.0.1 (127.0.0.1) 56(84) bytes of data.
    64 bytes from 127.0.0.1: icmp_seq=1 ttl=64 time=0.021 ms
    64 bytes from 127.0.0.1: icmp_seq=2 ttl=64 time=0.030 ms
    ^O64 bytes from 127.0.0.1: icmp_seq=3 ttl=64 time=0.026 ms
    64 bytes from 127.0.0.1: icmp_seq=4 ttl=64 time=0.049 ms
    ^C
    --- 127.0.0.1 ping statistics ---
    4 packets transmitted, 4 received, 0% packet loss, time 3079ms
    rtt min/avg/max/mdev = 0.021/0.031/0.049/0.010 ms

On macOS 15.3.2, with the Terminal app, it does work as expected.

    $ stty -a | grep 'discard\|flush'
    	-echoprt -altwerase -noflsh -tostop -flusho pendin -nokerninfo
    cchars: discard = ^O; dsusp = ^Y; eof = ^D; eol = <undef>
    $ ping 127.0.0.1
    PING 127.0.0.1 (127.0.0.1): 56 data bytes
    64 bytes from 127.0.0.1: icmp_seq=0 ttl=64 time=0.087 ms
    64 bytes from 127.0.0.1: icmp_seq=1 ttl=64 time=0.221 ms
    ^O^C
    --- 127.0.0.1 ping statistics ---
    9 packets transmitted, 9 packets received, 0.0% packet loss
    round-trip min/avg/max/stddev = 0.068/0.171/0.362/0.087 ms

### Other Stuff

My initial draft of this post had a few additional sections that have since been moved to their own posts:

- Multiple URLs in Git Remote
- Touch Typing Number Keys