A genius? I just love studying.

Chapter 85 I Can’t Sleep (Sixth Update)

brush!
After writing the last character, Fang Wen felt refreshed and relaxed all over.

he made it!

He solved the problem that had troubled him for a month in just one night!
I looked at the time on the computer, it was 3:17 in the morning.

Looking up and out the window, the campus was quiet, with only the dim light of the street lights casting an endless night.

Have you ever seen Chengdu University at three in the morning?
Fang Wen can proudly say that he has seen it!
Of course, this was not uncommon when he played games before.

But I have seen that the sense of accomplishment brought by staying up late to play games is simply incomparable to that of staying up late to write a paper at three in the morning.

After the excitement wore off, a deep sense of tiredness washed over him. Without even bothering to wash up, Fang Wen climbed onto the bed with his phone in hand.

However, what made Fang Wen regretful was that he pressed the lock screen button and checked the time before going to bed.

Then he saw the WeChat message on his phone.

It was sent by Chen Hui, just a few minutes ago.

"Doesn't he ever sleep?"

With this question in mind, Fang Wen opened the WeChat message.

"Your paper is very interesting, but I have some questions.

In step 2 of the proof, when the automorphism is represented as a matrix, the coefficient l is assumed to be compatible with the action of GL(1, p), but the necessity of l mod p being non-zero is not verified.

Is it possible to ensure that the commutator generates the minimum generator of Φ(G) by embedding the parameter constraint l≡0modp? By proof by contradiction, we show that if l≡0modp, then [a, b]∈Φ(G)p[a, b]∈Φ(G)p, resulting in a decrease in the nilpotent class, contradicting the original hypothesis.

We then construct an adjoint representation to verify the effect of the automorphism on the generator as a matrix and verify the consistency of matrix multiplication with group operations through direct calculation. In particular, the relationship b^p=a^(kp^n-2) must satisfy m≡1modp^(n2), thus restricting m∈GL(1,p).

"Ok?"

Fang Wen quickly jumped out of bed, turned on the computer again, opened his paper, and turned to step two of the proof process.

After spending more than ten minutes and with Chen Hui's reminder, he indeed discovered that there were indeed loopholes in his proof process, or at least it was not rigorous enough.

What's his brain like?

The difficulty of discovering a problem from scratch and discovering a problem after being reminded is incomparable.

This is the famous P vs. NP problem in mathematics.

For example, if you are at a banquet and want to find your high school classmate among the 345 people attending the banquet, it is very difficult. But if someone points to someone and asks you if that person is your high school classmate, it will be much easier.

This is the gap between him and Chen Hui now!

He is a graduate student!
Fang Wen's mind was a little confused, and he no longer dared to treat Chen Hui as a high school student.

I wonder if Director Liu noticed it when he read the paper?
Without thinking too much, he continued to think and calculate based on Chen Hui's solution. Half an hour later, Fang Wen's eyes were filled with bright light.

feasible!

The solution given by Chen Hui is completely feasible!

It can even be said to be very clever!

If the proof of the completeness of the Frattini quotient group improvement made this thesis complete, then resolving the coefficient compatibility vulnerability in matrix representations made it an excellent thesis. Fang Wen even fantasized about whether he could compete for next year's outstanding graduate award.
It turns out that it’s so cool to hang out with a big boss!

This is simply feeding the food into your own mouth!
He opened his phone again and saw that Chen Hui had sent him several messages, but he only read one.

"In step 3, we directly assert ker()Hom(G/Φ(G), Φ(G))ker()Hom(G/Φ(G), Φ(G)), but we do not prove that every homomorphism can be lifted to an automorphism. We need to construct a bijective mapping, ψ:Hom(G/Φ(G), Φ(G))→ker()ψ:Hom(G/Φ(G), Φ(G))→ker(), which is ψ(f)(a)=af(aΦ(G))ψ(f)(a)=af(aΦ(G)), ψ(f)(b)=bf(bΦ(G))ψ(f)(b)=bf(bΦ(G)), and verify that it is a monomorphism and surjective.

Then, using the theory of derivations, we regard the automorphism in ker()ker() as a derivation from G/Φ(G)G/Φ(G) to Φ(G)Φ(G), and cite the Hochschild-Serre spectral sequence to prove the triviality of H1(G/Φ(G), Φ(G))H1(G/Φ(G), Φ(G)), thus confirming the rationality of the bijection.

In step 4, it is assumed that the semidirect product exists naturally, but no explicit section is constructed to prove the splitting property, which may lead to the decomposition not being valid. We can use the power automorphism to construct the section and define the section s:GL(1, p)→Aut(G)s:GL(1, p)→Aut(G) as s(m)(a)=a^m, s(m)(b)=bs(m)(a)=^am, s(m)(b)=b, and verify its homomorphism and right inverse property with the quotient group mapping.

Then, through the cohomology cancellation argument...

Extreme parameter situations may lead to structural collapse, and the parameter thresholds can be discussed in a classified manner...increasing the strictness of the preconditions..."

. . .

Fang Wen felt a headache when he looked at the densely packed formulas on the screen. Was his paper really full of loopholes?
After a period of restlessness, Fang Wen calmed down again and carefully reviewed his paper again in accordance with Chen Hui's suggestions.

Time flies like water, and in the blink of an eye, the sky is already turning pale.

At six o'clock in the morning, Chen Hui woke up on time, raised his hands in front of his eyes, and opened his sore eyes with some difficulty.

Sure enough, the human body has its limits, and not a minute less than three hours is allowed.

I was so excited about reading the paper last night that I went to bed more than ten minutes late, and my body gave me a warning this morning.

But it's all worth it.

In less than five hours last night, my math proficiency increased by 5%!

It is equivalent to the results of his two weeks of study.

With his level of study intensity, the amount of content he learned in two weeks can be said to be enormous.

This also allowed Chen Hui to find the path to the future!
Of course, he also understood that just writing papers would not lead to such rapid improvement. The reason why he was able to make such rapid progress last night was mainly due to his accumulated experience.

If doing exercises is to consolidate the knowledge learned, then writing papers is to apply the knowledge learned. Before, he was only learning and consolidating, but lacked application.

In the process of application, it is equivalent to sorting out all the knowledge he has learned. Although no new knowledge is generated, he is now familiar with the knowledge context of group theory.

It was like he had a pile of cement and bricks before, but they were placed randomly, and now, they have built a solid foundation and become an iron plate!
This is why math proficiency has skyrocketed.

So learning is equally important, and the process of learning is the process of obtaining cement and bricks.

It seems that writing a paper should be put on the agenda!
However, I still have to participate in the finals of the Barbaria Mathematical Competition, so I should focus on learning and practicing problem-solving skills. This is equivalent to obtaining cement and bricks, and it is not a waste of time.

Wait until the finals are over, and then think about writing a paper based on what you have learned.

As for writing the thesis, there is a real graduate student in the mathematics department in his address book, and he can consult with him to determine the direction.

As expected, those who help others will be helped by others!

Chen Hui's mind became clearer and clearer, and the light in his eyes became brighter and brighter.

I cheered myself up, tidied the quilt, pulled up the sofa, opened the curtains in the living room, went to the kitchen to steam two buns, and then started to wash up.

(End of this chapter)