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Embedding Dihedral Groups in Vanishingly Small Symmetric Groups

16 Jul 2022 - Tags: sage , pretty-pictures

After the long and arduous process of writing my previous posts on homotopy theories and $\infty$-categories, it’s nice to go back to a relaxed post based on an mse question I answered the other day. Nature is healing ^_^.

This post was asking about the smallest set on which certain groups can faithfully act. I actually wrote about this same idea almost exactly a year ago in a blog post here, and while answering this question I remembered that I wanted to follow up on that post. I mentioned some asymptotic result at the end of it that’s super hard to use, and I was sure I could get a more legible result if I wanted to1.

Well, when answering this question I was reminded that I want to!

First, let’s recall what question we want to answer:

In the last post, we showed that the dihedral group of an $n$-gon’s symmetries \(D_{2n}\) can embed into the symmetric group \(\mathfrak{S}_m\) even if $m \lt n$. For instance, \(D_{2 \cdot 6} \hookrightarrow \mathfrak{S}_5\).

A perfectly natural question, then, is how much smaller can we make this?

We showed that if $n = \prod p_i^{k_i}$ is the prime factorization of $n$, then $D_{2n} \hookrightarrow \mathfrak{S}_{\sum p_i^{k_i}}$. For instance, since $6 = 2 \cdot 3$, we have \(D_{2 \cdot 6} \hookrightarrow \mathfrak{S}_{2+3} = \mathfrak{S}_5\).

With this in mind, it seems entirely believable that we can get the ratio to be very small.

As is usually the case, before trying to prove this, I wanted to compute some values and see how quickly things are decreasing. This wasn’t really necessary in hindsight, but it did make for some pretty pictures!

First, here’s a plot of the minimal $m$ so that $D_{2n} \hookrightarrow \mathfrak{S}_m$.

You can see that the maximal values are always $m=n$, which occurs at all the prime powers. However, you can also see that the minimal values can be substantially smaller.

To get a handle on just how much smaller, let’s plot the ratios $m/n$ instead.

As a (very) quick exercise, for which $n$ do we hit the ratio $1$? How often do these occur?

As a less quick exercise, set $N = 1000$ in the above code and notice how we stratify into multiple limiting ratios. Can you tell what some of these ratios are?

Lastly, let’s take the minimal ratio we’ve seen so far

I would love to find a nice curve upper bounding this last scatter plot, but it seems like a possibly tricky problem in number theory. Formally:


\[f(n) \triangleq \displaystyle \min_{\prod p_i^{k_i} \leq n} \frac{\sum p_i^{k_i}}{\prod p_i^{k_i}}\]

Can you find asymptotics for $f$?

If anyone wants to take a stab at this (or any other problems related to this sequence2), I would love to hear what you find ^_^.

Whatever the asymptotics are, it’s clear that $f(n) \to 0$. That is, for any $\epsilon$ you like, there’s a dihedral group $D_{2n}$ embedding into a symmetric group $\mathfrak{S}_{n \epsilon}$. I don’t know why, but this feels remarkable to me. It says that somehow we can get away with practically no objects at all in order to faithfully represent a dihedral group action. Said another way, there’s some $n$-gon whose symmetries can be faithfully represented in the symmetries of only $\frac{n}{1000000}$ many points.

For completeness, let’s give a quick proof of this fact. It’s fairly easy, so I encourage you to try it yourself! In fact, it’s quite easy to prove various strengthenings of this fact. For instance, the proof we’re about to give shows that we can take $n$ to be a product of $2$ primes, and is easily tweaked to allow us to put lots of bonus conditions on the prime factorization of $n$.

$\ulcorner$ Let $\epsilon \gt 0$.

Pick two primes $p,q \gt \frac{2}{\epsilon}$. Then from the results in the last post the dihedral group $D_{2pq}$ embeds into the symmetric group $\mathfrak{S}_{p+q}$. But now

\[\frac{m}{n} = \frac{p+q}{pq} = \frac{1}{q} + \frac{1}{p} \lt \epsilon\]

That is, $f(pq) \lt \epsilon$, so that $f(n) \to 0$. $\lrcorner$

It’s nice to write up a quick relaxing post for once. I thought about trying to answer some of the number theoretic questions that I posed throughout (as well as a few that I didn’t), but I really wanted to keep this quick. Plus, I suspect a lot of these questions will be somewhat simple, and I might keep them in my back pocket in case a younger undergraduate comes to me looking for a project.

Also there’s an extra reason to be excited about this post: it gave me an excuse to submit another sequence to the OEIS! The inputs for which $f(n)$ changes are interesting, and were not in the OEIS already, so I submitted them while I was writing this up! If all goes well, they should be available as A354424 in the near future3.

It’s also nice to go back to an older post and give it the closure it really deserves. The asymptotic result I cited there is borderline unusable, and this post answers the question that I really wanted to ask in that post4.

Take care, and stay safe all ^_^. Talk Soon!

  1. While writing that post all those months ago, I certainly did not want to. If I remmeber right, I finished writing that post at like 3 or 4 in the morning… 

  2. For instance, how long can the wait time be before we see a better ratio?

    Precisely, let $a_n$ be the sequence of values so that $f(a_n) \neq f(a_n-1)$. How far apart can $a_n$ be from $a_{n+1}$?

    Said another way, how large are the gaps in A354424

  3. I don’t know why, but submitting to the OEIS always makes me kind of giddy with excitement! 

  4. It also brought up a whole host of other questions! This is a huge part of the fun of math for me – there’s always more to explore.