On China

I’m talking to that 犹太IMO金牌 again. I first asked him if he knew the Riesz representation theorem, the statement of which I saw today. He said he used to. Then I brought up Shizuo Kakutani, who was quite a genius mathematician, who created some generalization of the aforementioned theorem or something like that. His daughter Michiko is also a distinguished writer. On that I said:

Lol I haven’t gotten to meet many Japanese
They don’t emigrate much nowadays, so patriotic
They’re so well organized and efficient
Produces lots of top mathematicians too

He responded with “china weak.” And “china deserved to get fucked by japan.”

On that, I was like:

China was super weak back then
Of course, the situation has reversed/is reversing
China is still behind Japan in many advanced areas, but it’s just a matter of time
Japan lost to America in WWII
China on the other hand could defeat America in the Korean War
Thanks to communist ideology

He said that “china did not defeat america.” I responded:

It was a stalemate whatever
But China proved it could get even with number one in the world
When she was still very behind
In any case, in the war in North Korea, America clearly lost, America had to flee
If China had better logistics and equipment probably could’ve taken over the entire Korean peninsula
Because of the Korean War, many of those top Chinese in STEM in America returned
There were negotiations as America knew if they let them return these people would serve their enemy
People contrast that to the brain drain after reforms
The younger generation of Chinese do not have the type of selfless patriotism that the older generation did
Lol you don’t like China
I think America lost its best chance to bring China down, that was during the 89 protests
That was actually kind of close
It’s quite remarkable that China recovered so well. When you’re down, it’s really hard to get back up.
In any case, by 1970s, people in China knew that the most difficult/critical period was past.
And that China had succeeded at it
It’s like earning money, the beginning is the hardest, once you’re rich and high up, it’s almost impossible to fail

He says: “fuck china. china anti human rights.” It’s funny how so many people say that, and I believe privately, or not so much, many in the world have a rather low opinion of China. Though I’m Chinese, I wouldn’t say I really care; it’s just a perception as far as I can tell, not something that can be objectively defined. When I grew up in America, I kept hearing this negative stuff about China and was wondering what was going on. Back a decade ago, China was much less developed than now, and perhaps because of that, the bashing sometimes feels to have subsided quite a bit now compared then, but maybe not, considering that even a guy like him will say that. Whether he genuinely believes it, that is another matter.

On this, I’ll give some of my thoughts. Recall that I said in my chat with him: “when you’re down, it’s really hard to get back up.” This is in general, it applies to individuals as well, with unemployment and such. In the context of the chat, I was referring to the century from 1850 to 1950, when China kept being beaten and made little progress when the rest of the world was advancing rapidly, including China’s foe from the East, Japan. Back then, many intellectuals desperate believed China to be hopeless and on that, even advocated the abolishment of Chinese characters. I believe China was very fortunate to get out of that, as it could have easily been much worse. The international situation, in particular the world’s having been exhausted after WWII destruction, gave China the opportunity to win the civil war, ending a century of violent internal strife that had severely hampered development. The Korean War did much to help Chinese regain their confidence. It proved Chinese military ability for the first time in modern history, much needed at the time, and America blundered by letting China do so. The 1950s was a golden period for China, during which with Soviet aid, China modernized essentially, developing the industrial foundation that even after the Soviet Union withdrew its support for China, though it had a significant negative effect development, China was able to do okay. In the 60s, the international situation was very unfavorable for China, but by 1970, China was high up enough in terms of capability that America had no choice but to recognize it, seeing that there was no way the old regime in Taiwan could retake the mainland. At that time, China was still extremely poor standard of living wise, but there was already a fair degree of technological sophistication. China was also very lucky not to suffer the demise that the Soviet Union did that is literally impossible to recover from. Why that did not happen, why America did not succeed in 1989 in bringing China down, is a very complex question. The Chinese elite were not as foolish as the Soviet ones. Since then, China has made tremendous progress in terms of developing economics and standard of living and also STEM, and though of course, China is still behind in certain areas, it is only a matter of time as many believe before the gap closes. Throughout the last 50 years, these “experts” have doubted the PRC, but the PRC keeps proving them wrong. Maybe these “experts” should stop deluding themselves on many matters.

I’ll give my personal opinion. I believe that every individual, every nation, should develop in a way that’s most suitable for them. Copying someone very high up blindly usually leads to disaster, because that high up person is genuinely well equipped enough to do what you are realistically unable to presently do. Instead, try to find something that hasn’t been tried before without large risks that you have an argument might be successful for you. This was exactly what China did and is still doing, so far to great success. China was lucky to switch its system at about the right time compatible with its circumstances. In the 1950, China was emerging from a century of war and stagnation, and the odds seemed so against her. Many could not believe that the communists would win the civil war, as ill equipped as they war. They did so, according to many, by developing a unique way of combat. It shocked the world that with it, China, avoiding its weaknesses, was able to succeed against the most powerful country in the world at war. China had no air force or navy at that time, with the exception of what of that had just been provided to China by the Soviet Union. China also made the right choice of using the Stalinist style economy that had already been proven to be successful in the Soviet Union, which enabled her to modernized very rapidly in a decade. China, as far as I can tell, had no intention to go to war with America and no expectation that it would. If not for MacArthur’s foolish and miscalculated decision to invade North Korea, China probably would have established normalized relations sooner or later and would not have leaned so one-sidedly towards the Soviet Union, and would not have taken such extreme measures at home subsequently. Another major success was that China was able to establish relations with America before Mao’s death on relatively good terms, owing to the capability China possessed at that time in addition to that threat posed to China by the USSR brought about common interest between the two. China did not stick to the old planned economy, instead embarking on a mixed economy with gradual proliferation of private enterprise, seeing perhaps that it was past the stage when the Stalinist style economy was needed. At the same time, China did not fall for the market fundamentalism that America has, and simultaneously, China kept its faith in developing a political-economic system suitable for the stage that it was at amidst enormous pressure, especially after the death of the USSR brought about a tide of “the end of history” in international political thought. Now China seems to be doing quite well, with rapid development economically and also in science and technology, and about to become competitive at the forefront in arenas at which she had been seen as backward. As this happens, Chinese, with a deeply engrained inferiority complex, becoming more confident in themselves and in their system, which with political bias aside has many advantages, such as long term planning.

It is interesting how many very intelligent people in the West, including the person I mentioned in this very post, believes some rather peculiar notions on China related matters. It still puzzles me where they’re coming from with all that. They can not like China or see China as a threatening competitor (and I won’t be offended by that, as people are entitled to their own view), but they should still try to be objective, as unpleasant as the facts may be for them to bear. Penalizing someone or downgrading someone’s ability or accomplishment out of an antipathy for that person’s background or political/religious beliefs is the act of a little person, an insecure person. Also, when you discriminate against someone and they still beat you, it’ll only make them more formidable and yourself more insecure.

Last but not least, I’ll reiterate again that Anglo culture is still dominant across the globe, as a legacy of British colonialism as well as subsequent American supremacy. With that said, international discourse will necessarily be biased towards the interests of that group, an obvious fact that apparently still needs to be noted, and a rationalist would apply some correction to account for the bias. On the other hand, Chinese language and culture is still alien to most of the world, and a derivative of that is that there is much vital information accessed little outside of China of much more validity than what the Anglo media chooses to promulgate. I know that there are ones keen on using such means to alter political opinion and whatnot, so as to bring down a regime they don’t like, as was done in Ukraine in 2014, but these are rogue tactics that will eventually reflect badly on its instigators. Plus, time and time again, Chinese have proved not foolish enough to fall for these tricks.

Various thoughts, here and there

Another exhausting week of work is past, and I am presented with another chance to wind down. Last night, through various casual reading, I was reminded of the concept of “effeminacy.” In many contexts, if you’re a man, the worst you can be seen as is “effeminate.” And of course, American culture stereotypes Asian men (specifically East Asian men) as effeminate. On this, there is this and this, among many other similar articles, especially the notorious this, which I stumbled on several years ago. I’ll say that there is as far as I can tell some truth to this from an objective biological point of view, in the likes of higher and softer voices and lower testosterone levels in East Asian men relative to white men, and also in white men relative to black men. On this note, I was also reminded of some comment of Michael O Church on reddit (which I cannot easily find anymore) that used “high IQ androgyny” as a factor to illustrate how super smart people (like +3.5 even +4 sigma g) get smashed in the corporate world, politically. It brought me to wonder if far tail g men really are more effeminate, which is likely to be the case, as there has got to be some biological tradeoff for the substantially larger brain that is the material source of such extreme cognitive ability. The light, nasal (or whatever you call it, for lack of better word I can think of) of voices of various mathematicians echoed back to me one after another, in contrast to the deeper and superficially more assertive and aggressive (and masculine) voices of those dumber business guys (mostly WASPs) in positions of power in the corporate world. Speaking of WASP, I could see also how Jews are perhaps more effeminate as well; I’d seen and heard enough that I feel I could intuitively recognize a Jewish voice as well, with of course there being the style of language in combination with the vocal mannerisms that sounds it more distinctively Jewish. I’ve long noticed that though East Asians reared in US mostly speak without an accent, it is still often easy to tell that it is an East Asian voice; such just goes to show how real race is, how rearing in an alien land and culture changes not the deeply engrained racial characteristics which go beyond physical appearance. It is quite a marvel indeed and a beautiful product of human evolution, a panorama of human biodiversity, that of course varies significantly more among individuals of groups than among the group averages. East Asians may be effeminate in the sense aforementioned, but interestingly, there is this “mad Asian guy” on Steve Hsu’s blog who in comments has written of the East Asian cognitive profile relative to the white one as akin to the male one vis-a-vis the female one, which is the undebatable truth born out by the result of testing that has across generations revealed East Asians to be higher by about if not at least two-thirds standard deviation in math and visual spatial, thereby making white people more effeminate in another sense.

Human (biological) development is quite fascinating, especially those of outliers, of which I am one to quite a degree, though probably not enough to make me genuinely distinguished (as in 1 in 100,000 or even 1,000,000 in terms of rarity of ability and accomplishment) in any way ever. We all know that top athletes in legit sports like Shaq are physical freaks of nature who deviate dramatically from the norm in their physical development but in one still constrained by the bell curve, physically speaking, but more consequentially, there are geniuses of mind the brains of which are extremely unusual in such a way that concrete output, in terms of quality and profundity, and to a lesser extent quantity is exponentially increasing as one goes further along the tail in our artificially created bell curve. These people, by virtue of their unusual brain structure, are able to perceive the world, engage in a form of consciousness, far beyond, and more objectively correct than, what the normal human experiences. One can only put oneself in another mind rationally; it is impossible to do so for real. As an example to illustrate, it is impossible to feel the way a mentally sick person does when one is healthy and that does not change for one who has recovered from a depressive episode with respect to oneself. Saying this brings to my mind the following quote of Huxley:

Perhaps men of genius are the only true men. In all the history of the race there have been only a few thousand real men. And the rest of us–what are we? Teachable animals. Without the help of the real man, we should have found out almost nothing at all. Almost all the ideas with which we are familiar could never have occurred to minds like ours.

This quote obviously has eugenicist overtones. It hints at a need of measure to protect against degeneration of a human species that for all its wonder and power, relative to real animals, is still mostly degenerate. It also evokes fear, especially in this day and age when genomic prediction and selection is imminent, that some elite will use it to rule an ignorant, subservient masses. We are and already have been in such a world for ages, with the heritability of ability coupled with transmission of money and power in a feudalistic manner, just now in disguise, with additional channels for mobility now for those of ability from lower background. The elite still rules and exploits the masses, just in a way more benign than ages ago with slavery and priestly spiritual opium abound. There is all the evidence that the best we can create as a society is to provide for everyone an environment and position and work appropriate for his ability and temperament taking into account of course the needs of society. There is also the observation that in general, those of higher ability have higher expectations in terms of what they do and also in terms of their material life, just as kids expect less than adults, in terms of their lifestyle. While there are significant differences in ability, and it probably wouldn’t hurt to make people a bit smarter, I would not say there is anything terribly particular one should be, and that everyone should try to find and bring out what is suitable for one’s ability and one’s circumstances, and hopefully something that is never seen before. I’ve lived long enough to be aware that every stage there is a unique personal challenge, and that one’s position relative to others does not necessarily affect one’s happiness much.

History has witnessed clashes, often in the form of war, between races, cultures, beliefs, systems. Groups have, in contempt or hatred of another, sought the other’s destruction, the other’s subjugation, the other’s cultural conversion, particularly in the religious sphere. It is human nature and also human weakness. There is a human inability to respect another far different from oneself for what the other is and a tendency in such cases to be imposing, with this’s being particularly prominent in certain religious cultures. I have had such myself, but they are as far as I can tell largely cleansed away, upon my realization of the differences in humans inherent and not permanently malleable, as evidence by significant changed in people once removed from parental pressure, once they are more free to do as they choose. In this regard, I take a more liberal attitude and come to cherish the diversity in culture, the diversity in styles of thinking, the diversity in talents across the world and across professions. I’ve come to realize over time that to feel contempt for another for his beliefs and tastes is futile, a waste of energy, and in such cases, parting ways, and viewing the difference as a mere reality, one observed and chuckled at, is the best way to go. Moreover, contempt can even metamorphize into some form of appreciation for the richness of our world in the variety it offers, in which one is but one constituent.


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, when 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 pth 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.

一说起偏微分方程,想到此行有不少杰出的浙江裔学者,最典型的可以说是谷超豪。想起,华盛顿大学一位做非交换代数几何的教授,浙江裔也,的儿子,曾经说起他们回国时谷超豪,复旦的,如他父亲一样,逝世了,又半开玩笑言:“据说谷超豪被选为院士,是因为他曾经当过地下党。”记得看到杨振宁对谷超豪有极高的评价,大大出于谷超豪在杨七十年代访问复旦的促动下解决了一系列有关于杨-米尔斯理论的数学问题。之外,还有林芳华,陈贵强,都是非常有名气的这套数学的教授,也都是浙江人。我们都知道浙江人是中国的犹太人,昨天Brian Bi还在说”there are four times more Zhejiangnese than Jews.” 可惜我不是浙江人,所以成为数学家可能希望不大了。:(



昨晚,我跟那位犹裔美国IMO金牌在脸书上讨论犹太人与中国人在最高智力层次相比的问题,想起有两位我所认识的以基督教传统长大的学习理论科学的美国人所我当时难以思议的东亚人智力上强于犹太人的观点。怎么说那,虽然在这前五十年,日本人和华人在理论科学上做出了的不少伟大的贡献,占有美国好研究大学不少教职,加上我这一代的华人在竞赛中出色的表现,可是还是感觉在科学里的绝顶,犹太人更多,以犹太人更具有一定的高瞻远瞩,可促以颠覆性的跨越,苏联那批犹裔数学大师为典型例子。同时,这个人,作为组合数学为学习及研究方向的高材生,又提醒我犹太人在理论计算机以及匈牙利式组合数学所有的牛耳。他说世界上最聪明的人是亚洲人,他的名字是Terry Tao,可是前一百犹太人综合强于前一百亚洲人的综合。对此,我问他:你了解任何Tao所做的工作吗,可肯定他是世界上最聪明的人?他回:我读过Green-Tao定理的证明。我没啥好说的,只言那还算比较前沿的东西,又跟他说我在对一些华罗庚撰的数论引导,虽引导,可以包含一些我现在认为相当深的数论,如Selberg所做的一些。Tao是个神,可是我也有朋友说:我有事想是否Tao未有过以自己不如von Neumann聪明而心里不安,加上数学那么难,连Tao都差点没有通过博士生资格考试。加上,von Neumann精通数门外语,具有即兴无迟钝翻译之能,以及过目不忘的记忆力,而我都看到过有些中国人在网上以将自己视为”primarily an Australian”的Tao对中国文化一无认同和他对中文一无所知表示反感。我在此博客上前所提到那位犹裔数学博士,念到深到Goro Shimura所做的工作,也觉得Tao有点overrated,觉得他的工作没有例如陈省身所做的深远及原创,说Tao至今还没有创造新的领域。关于犹亚之比,我想到的还有环境的因素,在这一点华人还是比较吃亏,由于经济原因,也由于名字及文化陌生原因,老一辈的华人还在为了自己及国家的生存挣扎,没有那么多经历投入科学研究。或许现在歧视对华人,即使在理论科学界,还是相当严重,虽理论科学少有集体性及宣传及政治因素,与比如生物或软件开发不同,可是人都是有偏见的,这包括评审委员会,如我听到的诺贝尔委员会对苏联科学家的工作的贬值。我这一代,华人在那些完全公平没有任何主观因素的竞赛里已经遥遥胜于犹太人,而那些是最好的对纯粹智力顶级的测试。我有时候想:中国人现在最缺的不是科学技术人才,而是反抗歧视,争取话语权的人才。在外国人眼中,中国人经常有性格被动的刻板印象,的确有这一点,但是好多也是不太客观的媒体所造成的。加上,中国人在美国也是少数,又有语言文化障碍,这又是一个视为寻常的Asian penalty.

数学上,我闻到了在\mathbb{F}_p域下的次数整除n的不可约首一多项式的积等于非常干净的x^{p^n} - x。此多项式很容易看到没有平方因式,用典型的此与此导数非共有因子去证。同时,取任意次数dd | n的不可约首一多项式\phi,则\mathbb{F}_p[x] / (\phi)是个p^d元素的域,则所有元素是x^{p^d} - x的根(x也是此域一元),从此可以得到任意多项式(这包括x)代到x^{p^d} - xx里都在模\phi等于零,也就是说他会是\phi的倍数。因d | nx^{p^d} - x | x^{p^n} - x,则\phi | x^{p^n} - x。不难证明\mathrm{gcd}(x^{p^n} - x, x^{p^d} - x) = x^{p^{\mathrm{gcd}(n, d)}} - x. 若d \nmid n,次数d的多项式若要整除x^{p^n} - x,必整除x^{p^{\mathrm{gcd}(n, d)}} - x,可以用归纳法证明此不可能,在\mathrm{gcd}(n, d)< d的情况下。从此,可以得到x^{p^n} - x没有因子次数非整除n。证闭。


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.

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.