## A observation on conjugate subgroups

Let $H$ and $H'$ be conjugate subgroups of $G$, that is, for some $g \in G$, $g^{-1}Hg = H'$. Equivalently, $HgH' = gH'$, which means there is some element of $G/H'$ such that under the action of $H$ on $G/H'$, its stabilizer subgroup is $H$, all of the group of the group action. Suppose $H$ is a $p$-group with index with respect to $G$ non-divisible by $p$. Then such a fully stabilized coset must exist by the following lemma.

If $H$ is a $p$-group that acts on $\Omega$, then $|\Omega| = |\Omega_0|\;(\mathrm{mod\;} p)$, where $\Omega_0$ is the subset of $\Omega$ of elements fully stabilized by $H$.

Its proof rests on the use orbit stabilizer theorem to vanish out orbits that are multiples of $p$.

This is the natural origin of the second Sylow theorem.

## 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.

## 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.

## 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.

## Galois group of x^10+x^5+1

This was a problem from an old qualifying exam, that I solved today, with a few pointers. First of all, is it reducible? It actually is. Note that $x^{15} - 1 = (x^5-1)(x^{10}+x^5+1) = (x^3-1)(x^{12} + x^9 + x^6 + x^3 + 1)$. $1 + x + x^2$, as a prime element of $\mathbb{Q}[x]$ that divides not $x^5-1$ must divide the polynomial, the Galois group of which we are looking for. The other factor of it corresponds to the multiplicative group of $\mathbb{F}_{15}$, which has $8$ elements. Seeing that it has $3$ elements of order $2$ and $4$ elements of order $4$ and is abelian, it must be $C_2 \times C_4$. Thus, the answer is $C_2 \times C_2 \times C_4$.

## More math

Last night, I learned, once more, the definition of absolute continuity. Formally, a function $f : X \to Y$‘s being absolutely continuous is its for any $\epsilon > 0$, having a $\delta > 0$ such that for any finite number of pairs of points $(x_k, y_k)$ with $\sum |x_k - y_k| < \delta$ implies $\sum |f(x_k) - f(y_k)| < \epsilon$. It is stronger than uniform continuity, a special case of it. I saw that it implied almost everywhere differentiability and is intimately related to the Radon-Nikodym derivative. A canonical example of a function not absolute continuous but uniformly continuous, to my learning last night afterwards, is the Cantor function, this wacky function still to be understood by myself.

I have no textbook on this or on anything measure theoretic, and though I could learn it from reading online, I thought I might as well buy a hard copy of Rudin that I can scribble over to assist my learning of this core material, as I do with the math textbooks I own. Then, it occurred to me to consult my math PhD student friend Oleg Olegovich on this, which I did through Skype this morning.

He explained very articulately absolute continuity as a statement on bounded variation. It’s like you take any set of measure less than $\delta$ and the total variation of that function on that set is no more than $\epsilon$. It is a guarantee of a stronger degree of tightness of the function than uniform continuity, which is violated by functions such as $x^2$ on reals, the continuity requirements of which increases indefinitely as one goes to infinity and is thereby not uniformly continuous.

Our conversation then drifted to some lighter topics, lasting in aggregate almost 2 hours. We talked jokingly about IQ and cultures and politics and national and ethnic stereotypes. In the end, he told me that введите общение meant “input message”, in the imperative, and gave me a helping hand with the plural genitive conjugation, specifically for “советские коммунистические песни”. Earlier this week, he asked me how to go about learning Chinese, for which I gave no good answer. I did, on this occasion, tell him that with all the assistance he’s provided me with my Russian learning, I could do reciprocally for Chinese, and then the two of us would become like Москва-Пекин, the lullaby of which I sang to him for laughs.

Back to math, he gave me the problem of proving that for any group $G$, a subgroup $H$ of index $p$, the smallest prime divisor of $|G|$, is normal. The proof is quite tricky. Note that the action of $G$ on $G / H$ induces a map $\rho : G \to S_p$, the kernel of which we call $N$. The image’s order, as a subgroup of $S_p$ must divide $p!$, and as an isomorphism of a quotient group of $G$ must divide $n$. Here is where the smallest prime divisor hypothesis is used. The greatest common divisor of $n$ and $p!$ cannot not $p$ or not $1$. It can’t be $1$ because not everything in $G$ is a self map on $H$. $N \leq H$ as everything in $N$ must take $H$ to itself, which only holds for elements of $H$. By that, $[G:N] \geq [G:H] = p$ which means $N = H$. The desired result thus follows from $NgH = gH$ for all $g \in G$.

Later on, I looked at some random linear algebra problems, such as proving that an invertible matrix $A$ is normal iff $A^*A^{-1}$ is unitary, and that the spectrum of $A^*$ is the complex conjugate of the spectrum of $A$, which can be shown via examination of $A^* - \lambda I$. Following that, I stumbled across some text involving minors of matrices, which reminded me of the definition of determinant, the most formal one of which is $\sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)\prod_{i=1}^{n}a_{i,\sigma_{i}}$. In school though we learn its computation via minors with alternating signs as one goes along. Well, why not relate the two formulas.

In this computation, we are partitioning based on the element that $1$ or any specific element of $[n] = \{1, 2, \ldots, n\}$, with a corresponding row in the matrix, maps to. How is the sign determined for each? Why does it alternate. Well, with the mapping for $1$ already determined in each case, it remains to determine the mapping for the remainder, $2$ through $n$. There are $(n-1)!$ of them, from $\{2, 3, \ldots, n\}$ to $[n] \setminus \sigma_1$. If we were to treat $1$ through $i-1$ as shifted up by one so as to make it a self map on $\{2, 3, \ldots, n\}$ then each entry in the sum of the determinant of the minor would have its sign as the sign of the number of two cycles between consecutive elements (which generate the symmetric group). Following that, we’d need to shift back down $\{2, 3, \ldots, i\}$, the presentation of which, in generator decomposition, would be $(i\ i+1)(i-1\ i) \ldots (1\ 2)$, which has sign equal to the sign of $i$, which is one minus the column we’re at, thereby explaining why we alternate, starting with positive.