As an overkill example, the happy family are subquotients of the monster group, but the monster group is simple.
Does not have any non-trivial normal subgroup.
And therefore, going back to our intuition that due to the fundamental theorem on homomorphisms there is one normal group per homomorphism, a simple group is one that has no non-trivial homomorphisms.
A Ring can be seen as a generalization of a field where:
- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that is not a ring because you can by addition reach the zero matrix.
- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't
So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)
A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.
And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.
Examples:
A similar model was used in the Hafele-Keating experiment to test special relativity on two planes flying in opposite directions. Miniaturization was key.
Contains a disposable tube with 6g of Caesium. You boil it, so when it runs out, you change the tube, 40k USD. Their tube is made by Agilent Technologies, so a replacement since that opened in 1999, and the original machine is from the 60s.
Detection is done with an electron multiplier.
youtu.be/eOti3kKWX-c?t=1166 They compare it with their 100 dollar GPS disciplined oscillator, since GPS satellites have atomic clocks in them.
This technique is crazy! It allows to both:You actually see discrete peaks at different minute counts on the other end.
- separate gaseous mixtures
- identify gaseous compounds
It is based on how much the gas interacts with the column.
Detection is usually done burning the sample to ionize it when it comes out, and then you measure the current produced.
Gas chromatography by Quick Biochemistry Basics (2019)
Source. There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as .
It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!
Note that the vector product does not have to be neither associative nor commutative.
Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples
- complex numbers, i.e. with complex number multiplication
- with the cross product
- quaternions, i.e. with the quaternion multiplication
An algebra over a field where division exists.
Notably, the octonions are not associative.
Old illustration of Zhu Bajie
. Source. Needs better identification: en.wikipedia.org/wiki/Talk:Zhu_Bajie#Illustration_date_wrong:_15h_century?The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.
The group operation is called elliptic curve point addition.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact