## Math sunday

I had a chill day thinking about math today without any pressure whatsoever. First I figured out, calculating inductively, that the order of $GL_n(\mathbb{F}_p)$ is $(p^n - 1)(p^n - p)(p^n - p^2)\cdots (p^n - p^{n-1})$. You calculate the number of $k$-tuples of column vectors linear independent and from there derive $p^k$ as the number of vectors that cannot be appended if linear independence is to be preserved. A Sylow $p$-group of that is the group of upper triangular matrices with ones on the diagonal, which has the order $p^{n(n-1)/2}$ that we want.

I also find the proof of the first Sylow theorem much easier to understand now, the inspiration of it. I had always remembered that the Sylow $p$-group we are looking for can be the stabilizer subgroup of some set of $p^k$ elements of the group where $p^k$ divides the order of the group. By the pigeonhole principle, there can be no more than $p^k$ elements in it. The part to prove that kept boggling my mind was the reverse inequality via orbits. It turns out that that can be viewed in a way that makes its logic feel much more natural than it did before, which like many a proof not understood, seems to spring out of the blue.

We wish to show that the number of times, letting $p^r$ be the largest $p$th power dividing $n$, that the order of some orbit is divided by $p$ is no more than $r-k$. To do that it suffices to show that the sum of the orders of the orbits, $\binom{n}{p^k}$ is divided by $p$ no more than that many times. To show that is very mechanical. Write out as $m\displaystyle\prod_{j = 1}^{p^k-1} \frac{p^k m - j}{p^k - j}$ and divide out each element of the product on both the numerator and denominator by $p$ to the number of times $j$ divides it. With this, the denominator of the product is not a multiple of $p$, which means the number of times $p$ divides the sum of the orders of the orbits is the number of times it divides $m$, which is $r-k$.

Following this, Brian Bi told me about this problem, starred in Artin, which means it was considered by the author to be difficult, that he was stuck on. To my great surprise, I managed to solve it under half an hour. The problem is:

Let $H$ be a proper subgroup of a finite group $G$. Prove that the conjugate subgroups of $H$ don’t cover $G$.

For this, I remembered the relation $|G| = |N(H)||Cl(H)|$, where $Cl(H)$ denotes the number of conjugate subgroups of $H$, which is a special case of the orbit-stabilizer theorem, as conjugation is a group action after all. With this, given that $|N(H)| \geq |H|$ and that conjugate subgroups share the identity, the union of them has less than $|G|$ elements.

I remember Jonah Sinick’s once saying that finite group theory is one of the most g-loaded parts of math. I’m not sure what his rationale is for that exactly. I’ll say that I have a taste for finite group theory though I can’t say I’m a freak at it, unlike Aschbacher, but I guess I’m not bad at it either. Sure, it requires some form of pattern recognition and abstraction visualization that is not so loaded on the prior knowledge front. Brian Bi keeps telling me about how hard finite group theory is, relative to the continuous version of group theory, the Lie groups, which I know next to nothing about at present.

Oleg Olegovich, who told me today that he had proved “some generalization of something to semi-simple groups,” but needs a bit more to earn the label of Permanent Head Damage, suggested upon my asking him what he considers as good mathematics that I look into Arnold’s classic on classical mechanics, which was first to come to mind on his response of “stuff that is geometric and springs out of classical mechanics.” I found a PDF of it online and browsed through it but did not feel it was that tasteful, perhaps because I’m been a bit immersed lately in the number theoretic and abstract algebraic side of math that intersects not with physics, though I had before an inclination towards more physicsy math. I thought of possibly learning PDEs and some physics as a byproduct of it, but I’m also worried about lack of focus. Maybe eventually I can do that casually without having to try too hard as I have done lately for number theory. At least, I have not the right combination of brainpower and interest sufficient for that in my current state of mind.

## The Asian penalty

We all know that elite schools in the US discriminate against Asian applicants, essentially imposing a penalty for being Asian. And they have been rather pathetically pretending that such is not the case in spite of all the statistical evidence to the contrary. On this, people have said things like: where is affirmative action for Asians in the NBA/NFL? Well, today one of my colleagues who is a keen baseball fan, and probably also an NBA one, was talking about how there is even an Asian penalty in the NBA. Like, Asians are typically under-drafted, which means their number or rank in the draft is under commensurate with their actual ability and value at basketball. He says it’s due to the negative perception being Asian is for basketball in terms of how good one is yada yada yada. I haven’t paid attention to basketball for a long time, but I do remember the Linsanity several years ago, and when I was a kid, one who was a keen NBA fan, all the talk about Yao Ming. Last I checked Asians account for 0.2% of the NBA players, which means just a few names. Well, there is the height disadvantage after all. This was actually somewhat surprising to me, perhaps influenced by the fact that Yao Ming seemed to be overvalued due to the money he would bring from all the Chinese fans. So not only is there not affirmative action for Asians in the NBA, there is the same discrimination, the same penalty, the same stereotypes against Asians as in college admissions. Now that really sucks!

Now to something else that saddens me greatly that is a consequence of the current discriminatory policies against Asians in college admissions. Some Asian-Americans are afraid to check Asian and even afraid to engage in activities/pursuits they have gift and passion for, or at least some intrinsic interest in, under the fear that those are too Asian. Some are even afraid to show their Asian heritage and even reject their roots, which is quite sad, as you are who you are, in terms of your cultural background and denying it mostly makes you look quite pathetic. At least based on what I experienced growing up Asian in the states, many if not most Asian kids, even smart ones, try to distance themselves from their parents’ culture and are reluctant to learn or speak their parents’ native language, under social pressures osmosed in them by the whole American public school experience. Chinese culture is a pretty fucking cool and rich culture, with a beautiful language of artistic virtue that comes with a rich history. It is a pity that it is so misunderstood and that the American education system pressures against it in those from that cultural background. This is anti-intellectual in fact too, ironic as it is instigated partly by elite educational institutions, as reading multiple languages makes one’s mental world and whole spiritual existence a hell of a lot more interesting, an inevitable product of access of more diverse information.

To sum it up, it looks like all across the board America treats Asians as second class citizens. Yes, Asians are mostly new immigrants, but this is in fact overstated. As early as in the 19th century, Chinese in America were made to do much of the most dangerous work building the Transcontinental Railroad only to suffer the Chinese Exclusion Act. In the 20th century especially later on, Chinese as well as Indians, in addition to Japanese, Koreans, and Vietnamese, have created tremendous wealth for America, largely in science and engineering fields. Asians in America have mostly been busy working, busy creating, and some in hard menial labor in wretched conditions, with such being a major contributor to Asians, as a group, being seen as passive and apolitical, creating a self-pertuating stereotype. On this, Steve Hsu has expressed on his blog how slow the Asian community has been to organize against the double standards imposed on them in college applications, relative to the what Jewish community had done when the same had been unjustly instituted against them. Given the voluminous extent to which Asians as a group have contributed to America in terms of innovation and value creation, Asians have every right to demand that they are fairly considered for all positions, and leadership positions in particular, based on merit, which is not happening right now. On this more Asian-Americans ought to muster the courage to speak up for what is right, as Steve Hsu, Yukong Zhao, and Jian Li, among many others, have done amidst resistance.

## Asymptotic formula for square free integers

\begin{aligned} \displaystyle\sum_{n \leq x} \displaystyle\sum_{d^2 | n} \mu(d) & = \displaystyle\sum_{d \leq \sqrt{x}} \mu(d)\left\lfloor \frac{x}{d^2} \right\rfloor \\ & = x\displaystyle\sum_{d \leq \sqrt{x}} \frac{\mu(d)}{d^2} + O(\sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(x\displaystyle\sum_{d > \sqrt{x}} \frac{1}{d^2} + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O((1 + \sqrt{x}) + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(\sqrt{x}). \end{aligned}

## Composition series

My friend after some time in industry is back in school, currently taking graduate algebra. I was today looking at one of his homework and in particular, I thought about and worked out one of the problems, which is to prove the uniqueness part of the Jordan-Hölder theorem. Formally, if $G$ is a finite group and

$1 = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_r = G$ and $1 = N_0' \trianglelefteq N_1' \trianglelefteq \cdots \trianglelefteq N_s' = G$

are composition series of $G$, then $r = s$ and there exists $\sigma \in S_r$ and isomorphisms $N_{i+1} / N_i \cong N_{\sigma(i)+1} / N_{\sigma(i)}$.

Suppose WLOG that $s \geq r$ and as a base case $s = 2$. Then clearly, $s = r$ and if $N_1 \neq N_1'$, $N_1 \cap N_1' = 1$. $N_1 N_1' = G$ must hold as it is normal in $G$. Now, remember there is a theorem which states that if $H, K$ are normal subgroups of $G = HK$ with $H \cap K = 1$, then $G \cong H \times K$. (This follows from $(hkh^{-1})k^{-1} = h(kh^{-1}k^{-1})$, which shows the commutator to be the identity). Thus there are no other normal proper subgroups other than $H$ and $K$.

For the inductive step, take $H = N_{r-1} \cap N_{s-1}'$. By the second isomorphism theorem, $N_{r-1} / H \cong G / N_{s-1}'$. Take any composition series for $H$ to construct another for $G$ via $N_{r-1}$. This shows on application of the inductive hypothesis that $r = s$. One can do the same for $N_{s-1}'$. With both our composition series linked to two intermediary ones that differ only between $G$ and the common $H$ with factors swapped in between those two, our induction proof completes.

## A derivation of a Riemann zeta function identity

Yesterday, I saw the following Riemann zeta function identity:

$\displaystyle\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}$.

I took some time to try to derive it myself and to my great pleasure, I succeeded.

Eventually, I realized that it suffices to show that

$\{(dd_a, dd_b, d^2 n) : d_a | n, d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

and

$\{(dd_a, dd_b, n) : dd_a d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

are equal as multisets. As sets, they are both representations of the set of $3$-tuples of positive integers such that the third is a multiple of the least common multiple of the first two. In the latter one, the frequency of $(a,b,c)$ is the number of $d$ that divides both $a$ and $b$ such that $ab | cd$. In the other one, if we write $(a,b,c)$ as $(d_1 d_2 a', d_1 d_2 b', c)$ where $\mathrm{gcd}(a', b') = 1$, the $ab | cd$ condition equates to $d_1^2 d_2 a'b' | c$, which corresponds to the number of $d_1$ dividing $a$ and $b$ and such that $d_1^2 | c$ and with that, $d_2a', d_2b'$ both dividing $d_1^2 / c$, which is the frequency of $(a,b,c)$ via the former representation.

The coefficients $\{a_n\}$ of the Dirichlet series of the LHS of that identity can be decomposed as follows:

$a_n = \displaystyle\sum_{d^2 | n, d_a | \frac{n}{d^2}, d_b | \frac{n}{d^2}} (dd_a)^a (dd_b)^b$.

The coefficients $\{b_n\}$ of the Dirichlet series of the RHS of that identity are

$b_n = \displaystyle\sum_{dd_a d_b | n} (dd_a)^a (dd_b)^b$.

Observe how both are equivalent in that via the multiset equivalence proved above, $n$ determines the same multiset of $(dd_a, dd_b)$ for both and across that, the values of the same function $(dd_a)^a (dd_b)^b$ are summed. Hence the two series are equal.

## Automorphisms of quaternion group

I learned this morning from Brian Bi that the automorphism group of the quaternion group is in fact $S_4$. Why? The quaternion group is generated by any two of $i,j,k$ all of which have order $4$. $\pm i, \pm j, \pm k$ correspond to the six faces of a cube. Remember that the symmetries orientation preserving of cube form $S_4$ with the objects permuted the space diagonals. Now what do the space diagonals correspond to? Triplet bases $(i,j,k), (-i,j,-k), (j,i,-k), (-j,i,k)$, which correspond to four different corners of the cube, no two of which are joined by a space diagonal. We send both our generators $i,j$ to two of $\pm i, \pm j, \pm k$; there are $6\cdot 4 = 24$ choices. There are by the same logic $24$ triplets $(x,y,z)$ of quaternions such that $xy = z$. We define an equivalence relation with $(x,y,z) \sim (-x,-y,z)$ and $(x,y,z) \sim (y,z,x) \sim (z,x,y)$ that is such that if two elements are in the same equivalence class, then results of the application of any automorphism on those two elements will be as well. Furthermore, no two classes are mapped to the same class. Combined, this shows that every automorphism is a bijection on the equivalence classes.

## A recurrence relation

I noticed that

$(x_1 - x_k)\displaystyle\sum_{i_1+\cdots+i_k=n} x_1^{i_1}\cdots x_k^{i_k} = \displaystyle\sum_{i_1+\cdots+i_{k-1}=n+1} x_1^{i_1}\cdots x_{k-1}^{i_{k-1}} - \displaystyle\sum_{i_2+\cdots+i_k=n+1} x_2^{i_2}\cdots x_k^{i_k}.$

In the difference on the RHS, it is apparent that terms without $x_1$ or $x_k$ will vanish. Thus, all the negative terms which are not cancelled out have a $x_k$ and all such positive terms have a $x_1$. Combinatorially, all terms of degree $n+1$ with $x_k$ can be generated by multiplying $x_k$ on all terms of degree $n$. Analogous holds for the positive terms. The terms with only $x_1$ and $x_k$ are cancelled out with the exception of the $x_1^{n+1} - x_k^{n+1}$ that remains.

This recurrence appears in calculation of the determinant of the Vandermonde matrix.

## 两首诗

### Чанша

В день осенний, холодный
Я стою над рекой многоводной,
Над текущим на север Сянцзяном.
Вижу горы и рощи в наряде багряном,
Изумрудные воды прозрачной реки,
По которой рыбачьи снуют челноки.
Вижу: сокол взмывает стрелой к небосводу,
Рыба в мелкой воде промелькнула, как тень.
Всё живое стремится сейчас на свободу
В этот ясный, подёрнутый инеем день.
Увидав многоцветный простор пред собою,
Что теряется где-то во мгле,
Задаёшься вопросом: кто правит судьбою
Всех живых на бескрайной земле?
Мне припомнились дни отдалённой весны,
Те друзья, с кем учился я в школе…
Все мы были в то время бодры и сильны
И мечтали о будущей воле.
По-студенчески, с жаром мы споры вели
О вселенной, о судьбах родимой земли
И стихами во время досуга
Вдохновляли на подвиг друг друга.
В откровенных беседах своих молодёжь
Не щадила тогдашних надменных вельмож.
Наши лодки неслись всем ветрам вопреки,
Но в пути задержали нас волны реки…