Ultimate Recipe to Construct Every Finite Group - Group Extensions and Computational Group Theory
ฝัง
- เผยแพร่เมื่อ 16 พ.ค. 2024
- Group is a mathematical abstraction used to describe symmetry. In this video, we introduce the theory of groups, group decomposition and extension, and computational aspects as to why classifying all finite groups is impossible.
Links:
Video that introduces sporadic groups
Why Do Sporadic Groups Exist?: • Why Do Sporadic Groups...
Video that explains both algebraic and topological aspect of the spin group
Dirac's belt trick, Topology, and Spin ½ particles: • Dirac's belt trick, To...
Video for understanding the geometry of unit quaternions
Visualizing quaternions (4d numbers) with stereographic projection: • Visualizing quaternion...
Video that constructs the spin group as SU(2) axiomatically
The biggest misconception about spin 1/2: • The biggest misconcept...
Entire video series on spinors
Spinors for Beginners: • Spinors for Beginners
Eilenberg and Maclane's original paper on group extensions with non-Abelian normal subgroup
Group Extensions with a non-Abelian Kernel: shorturl.at/ryFLZ
And here are some resources to learn more about group extension:
ncatlab: ncatlab.org/nlab/show/group+e...
groupprops: groupprops.subwiki.org/wiki/G...
Chapters:
00:00 Intro
01:21 Examples of Groups
02:50 Equivalent Groups
05:32 Abstract Definition of a Group
07:22 Definition of Isomorphism
08:24 Automorphism Group
10:28 Decomposition of Groups
14:48 Finite Simple Groups
17:07 Infinite Group Cannot Be Factored
18:33 Group Extension
19:18 Direct Product / Classification of Abelian Groups
24:14 Semidirect Product
26:51 Non-split Extensions
31:45 What is Cohomology
35:37 Why Classification of Groups is So Hard
36:32 Isomorphism Testing Algorithm
39:02 Status on Classification of Groups
39:46 Classification of Groups of Order 16
42:42 Asymptotic Group Theory
43:32 Chemistry Analogy
44:45 Outro
Music🎵:
No.3 Morning Folk Song - Esther Abrami • No.3 Morning Folk Song
Corrections:
0:37 60 = 2 × 2 × 3 × 5
I feel like I need all of that information stretched out across a whole course worth of videos. This was a lot.
You know what they say about eating an elephant...one bite at a time. Yeah the video is a bit condensed...I'm on my 3rd viewing. It's kinda like a difficult chapter in a book. Just gotta keep watching until you understand, then move on to the next thing yuh don't, then tackle that.
@@jaytravis2487 i mean this covers a lot of material i learned or briefly touched on in a few of my modules in 3rd and 4th year of my masters in maths and then some. i dont really think this video is a great intro to group theory lol. the basics of group theory arent too hard but there's a lot going on here. i mean the first 20 minutes of this video is a fly-by overview of 2 algebraic structures courses, 2 group theory courses and a galois theory course, pretty much.
It consistently blows my mind that such rich and fascinating structures arise from the short and simple group axioms. Group theory is truly one of the marvels of the mathematical universe.
If I could have my time over, I would be a Group Theorist for sure.
I can barely put to words how pleased I am with this video. It touched so many interesting topics in group theory, summarized tons of info without disregarding rigor, and it shed a lot of light on many things that I learned in uni but didn't really developed a good intuition or could fully grasp their implications and relations to other parts of mathematics. Excellent work, I eagerly await for so many of the other topics you mentioned here that might get a video in the future.
😮
Your command and articulation of the material is highly impressive -- thank you for sharing your beautiful gifts!
This is a very good video. Of course, much has been skimmed. But it was very useful.
Basically a full semester course in group theory in a less than 1 hour video was very ambitious. And you pulled it off successfully without losing the attention of viewers.
Congragulations🎉
So, I just discovered a serious channel for math!! well done man! please keep up!!
Thank you for that introduction to cohomology! 👍
What an awesome video! Looking forward to more!
I have barely touched group theory and I can just barely put the pieces together but I found it heavily informative even if nearly disassociated with my knowledge
Very cool video! I would love to see more like it! Group extension theory and the theory of simple groups is very interesting to me!
Im taking a course at the moment on group ring and field theory, and this video sheds light on so many things I encountered that were confusing to me, and makes them seem far more intuitive. Great video!!!
Thank you for enlightening us, wonderful work.
there are definitely a few points made in this video that perhaps could have done with a proof - it's definitely not obvious that all abelian groups are either some cyclic group or a direct product thereof. Although, given how long it is, that definitely makes sense. It's always surprisingly hard to edit a maths video! Great job on this one, I look forward to seeing what else you come out with.
Great video! Abstract algebra is such a beautiful, interesting, and satisfying subject.
A good compilation of many group-related concepts. I'm not a mathematician; I was exposed to a variety of those concepts elsewhere but never imagined how they related to each other.
I cannot say I've learned all that has been exposed in this video. Some parts are very abstract. But at least I've been able to follow it all to the end, and it is an exponent of the high pedagogical level of this channel. Congratulations!
Excellent video and underrated channel!
What an amazing video!! It was a lot to take in though!
Such an amazing video! Wish I had this when studying Group Theory 😅
Hi. Nice video! However, I hope there were segmentations helping somewhat-experienced viewers to skip segments about topics they already knew. For example, I am familiar with groups in a basic level, but I'm still curious about group extensions and their classifications. I think there are a lot of viewers like me. Thanks!
loved it!
Thanks. I really enjoyed the video.
amazing thank you so much for this video, this is such a great visually supported explanation of a very abstract field and giving a strong intuitive overview of the way groups can be treated and "viewed" in that sense. Can you make a video about the simple lie groups specifically and go further into the differences of the periodic table?
I would love to see more of these connections between different groups and generators!
Do you have some literature recommendations for these visual intuitions on quotient groups etc. ?
Thank you so much for making this video. It has made me so much aware about the groups I see in physics. Please make more videos about the cohomology group.
i love this video, thank you!
Excellent !
I increasingly like these unorthodox tutorials. Or I’d call it speed-run math. The formal and strict old fashioned way often fails to deliver a point, an outlook, an overview, or even a warning - there be dragons.
I mean, in usual undergraduate class, group theory seems tamed - not hard, just abstract while being notation-wise and calculation-wise densely packed. With limited example (that can fit in the text and exercises) it gives a mirage that perhaps some sophisticated computer automation would be able to resolve any group’s structure. However this video showed it’s essentially a hard problem, hard not unlike P?=NP. The madness of higher-ordered exotic non-simple group is like a crazy house somewhere stashed in the backroom I guess.
Thank you so much
Amazing video!
10/10. I've been studying this topic and I was looking for something with a far sighted perspective to motivate me. This is very well done.
well done!
Very nice! I think I'll need to sit down and work through some of that again. Thank you!
33:20 I'd never heard cohomology described that way. Makes a lot of sense.
amazing video
Amazing
I remember how they used different letters for angles in calculus 3 and electromagnetism in physics, and it was very confusing for no reason
Different labels and nomenclatures like different languages hinder learning and understanding, but there isn’t enough return to adopt new standards, and like different languages there is subtle insight in how ideas are expressed. Maybe in a few centuries we will have cleaned up and standardized stuff.
@@chrislubs1341Importantly they enfasize the things you more comonly use in the respective fields, so there can actually be negative return on some type of standarizations.
so excellent, so beautiful
with regards
3am rabbithole spawned on my feed. bless the algorhythm. 41 minutes in and idk what i just watched
Great video, thanks a lot ! Would definitely watch a ton more of this fast paced, easy on the reminders, never seen on TH-cam, videos ! mmmmhhh yummy :P
Hey I know you kinda got some hate in the comments for this video because it was a little advanced, but I am very interested in this sort of thing! Like seriously this video is the only one of its kind explaining the theory of group extensions and frankly I’m all about it. I would love to see a video maybe explaining more about group structures and algorithms for understanding group theory.
For example I think the whole rank problem (IE given an arbitrary presentation find the size of the smallest generating set for the groups) for finite groups is fascinating. On paper it seems like a simple problem. You don’t even have to find the actual generators, you just have to guess at the number, and yet still it’s (as far as I know) Exp hard.
90-degree is not plural since it's an adjective
I love the vid
2:31 lmaoo
Amazing video! Thank you 😊
Would you mind making a video about applied character theory and representation theory?
Very interesting! I somehow always ignored the fact that there are other group products...
There's another context where the Pauli Group shows up, which is as the Cayley table of Cl(3). And as was shown, this group has the Quaternion group as a sub-group, which when added with ℝeal coefficients is isomorphic to Spin(3). More generally, Cl(n) has a very simple way to generate Spin(n) (as well as Pin(n), which is like Spin(n), but isn't "Special" since it includes reflections, so it doesn't have an S). I don't know if there's any connection between Spin(n) and SU(m) outside of the one instance of Spin(3) and SU(2), though I think Spin(n) might always be the double-cover of SO(n). (Using Clifford Algebras in computer graphics gives an embarrassingly simple way of interpreting the double cover, which is distinguishing between clockwise and counter-clockwise rotations. Ie, a 90° rotation clockwise is a separate rotation from a 270° rotation counter-clockwise, though since SO(n) only records the final position, they appear the same.)
Apologies for the tangent, but this was a really cool video.
Now I'm curious if there's a way to factor arbitrary Clifford algebras Cl(n) into how they split into C₂s.
Wow the animation for the segment on group cohomology was very cool (the first time I felt like I could "feel" what group extensions/cohomology was, however vaguely). Is it possible to develop these visualizations further? It seems like your visualization works for any sort of "product" of 2 cyclic groups. What about 3 cyclic groups? More general groups than just cyclic ones? I would really like to see more! Even just spelling how how to go between these "hands on" demonstrations to the abstract computation of the 2nd cohomology group.
LES GOOOOOOOOO
The generators of a group G are k ≤ log_2(|G|). This looks vaguely like entropy. Assuming the group operation is addition, the elements of the group G as combinations of generators k are at most the Shannon entropy of bits. Assuming the generators are logarithmic maybe finite groups have entropy?
Restricting to compact groups such as profinite and finite groups and looking at asymptotic group theory, it seems likely as profinite groups are probability spaces. Entropy can be defined using the statistics of probability.
very interesting observation. generators and their relations contain all the information of a group
your videos fucking slap dude keep it up lovemuffin
While it's good to maintain pace in a TH-cam video, this one could use an accompanying paper / Google Drive doc for the finer details. For example when defining isomorphisms you gloss over the fact that the group operation on the left is of the first group and the group operation on the right is of the second group. When giving an example of an isomorphism you simply write an arrow between SO(2) and U(1) without giving any rigorous definition of this arrow in terms of the matrix elements or the components of the complex numbers.
You can just look it up in a book
I don't know what you just said but it seemed cool
Love the SAO diss😂
Which are harder to classify, groups or knots?
Simple groups are easier than prime knots. Classification of Prime knots is far from complete, most likely impossible.
On the other hand, combining any 2 knot is rather easy concept, similar level to prime factorization of numbers. Grps, its the extension that is the hard part
Am I the only one not to like the common definition of groups as symmetries of an object ? I kinda see it the other way : symmetries of an object is a group
I like groups as “the invertible actions leaving the stable the input set”, symmetries do happen to be invertible, but I don't know why my brain does not like that way of talking about groups.
I might / must be wrong as it is the standard interpretation of groups, so feel free to correct me or suggest me something that can convince me
I kind of agree. "actions that leave an object looking the same" to just means doing nothing at all. It's much easier for me to understand when there's a non-symmetric thing that the group actions are acting upon, so that each distinct element of the group has a visually distinct object associated with it in a 1-1 mapping.
The three rules are also pretty easy to understand if phrased right. Associativity just means actions can be composed, invertibility just means you can undo an action, and identity just means that you have the option of doing nothing. This also makes more sense to me than talking about symmetries.
Infinity is not so much a tunnel as an ocean because the sequential differences among prime integers have a finite limit.
its the equivalent of
you found all the elements now do all of chemistry
But why Euclid argument doesn't work for simple groups though? Why we can't make some more simple groups by adding something to a multiple of existent simple groups?
Well, there are two problems here. Firstly, we can "multiply" groups (as in this video) by group extensions. However, we cannot add two groups with orders a and b together to get a group with order a+b; this really could not make any sense in the context of group theory. There is really no connection between the groups of order a and b, and the groups of order a+b. It's purely a multiplicative thing, and even there it's very complicated.
Secondly, you seem to have missed the fact that there *are* infinitely many simple groups. For example, in this video, it is explained that all of the prime cyclic groups are simple. There are infinitely many primes, so there are infinitely many prime cyclic groups. Not only that, but there are several infinite families of simple groups. The second one in this video is the alternating groups, which are simple for all A_n with n at least 5. It just so happens that there are also a finite handful of extra ones that don't fit neatly into any of these infinite categories, because they are too small for the large scale effects to take over and make them all have the same structure.
@@stanleydodds9 the note to the second one - I phrased it badly. pick all kinds of simple groups, multiply and "add one" like with Euclids proof to make it irreducable/simple. Current categories states that this kind of group will be homomorphic to one of existing simple group. Or is simply impossible to construct simple group from multiplication plus something ?
@@DeathSugarwell as they said, there is really no equivalent to "plus one" there. A group of order n and a group of order n+1 have nothing in common. You can say that the group of order n+1 doesn't have a group of order n (or any of its subgroups) as a subgroup (since n doesn't divide n+1) but that's not enough to conclude that the group of order n+1 is simple.
Moreover, Euclid's proof just shows that there are infinitely many primes, and that already implies that there are infinitely many simple groups, since Cp is simple for any prime p.
Sometimes, i live piecefully without knowing what horrible unordered mess waits me if i so happend to think about how to represent in code 3d axis transformation, then i jump straight into this paddle not expecting it to be deep and just drowning. Forgetting why i am here in the first place.
2:33 >abomination
what did he mean by this?
You know exactly what he meant.
@@Ivernet8319 ;-)
it's a shit anime
1:35 That group is called D4. That's objectively the only reasonable name for it. It acts on 4 elements, and that's where the 4 comes from. You wouldn't ever see someone insist that S24 has 24 elements, so arguing that D4 should be called D8 because it has 8 elements is inconsistent, which is bad. It is also plain silly to purposefully not use the odd numbers. I will take this fight every chance I get.
Yea you are right, imagine if somebody was talking about D_390 or some other random number and you would have to guess the shape they are reffering to
2
We just finished a basic introduction to group theory in college, during the lecture some examples were given and I asked myself the question, of can't these groups just be factored? Now I have the an answer, thank you for this insight into what lies beyond the basics.
2³ * 3 * 5 ≠ 60
You are wrong, there is a way to classify all finite groups using socle, the product of all minimal normal subgroups which is uniquely determined. So for each non simple finite group, it is an extension of socle, which is itself a direct product of simple groups, with smaller group, and the extension is classified by homomorphism into Out(socle) of that smaller group, where it is classified by H2 group with coefficients in center of Socle (product of commutative prime factors), if H3 conditionnis satisfied, though one needs to check that socle stays socle in extened group for each such extension to get uniqueness. This is a standard, canonical and unique way to build all finite groups, and the extension problem was indeed solved by papers you quote, but to get canonical representation it is indeed a small step.
Let me check if I’m understanding your statement correctly.
You are saying that, every finite non-simple group can be uniquely expressed as an extension of its socle, by a group which is smaller than that socle,
and that, given such a smaller group, the extensions of the socle of the original group by the smaller group, are classified by homomorphisms from the smaller group to the group of outer automorphisms of the socle (err.. I mean, the quotient of the group of automorphisms by its subgroup of inner automorphisms)
...and that this is the same as(?) the second group homology group of the smaller group (?) with coefficients in the socle of the group?
I didn’t understand the part about H3 condition.
How close was I?
@@drdca8263 Yes, extensions of K by G are classified by action of G on K (coming forn conjugation in extended group, you need to consider classes of homomorphisms containing inner homomorphisms of K, because G corresponfs to cosets in extended group) and then for each such action H2(G,Z(K)) where Z(K) is center of K considered as a ring with that action, and exist when H3(G, Z(K)) is zero. That is the content of that paper from WW2 era that he quotes. So all you need to do to get all groups is to find some canonical normal subgrouop of each group, and one possibility is socle. This his how groups are indeed considered, number of such socle kernels in socle subnormal series which is unique for each group is class of group and is well defined. Non-isomorphic groups can be listed systematically in this way (unlike what this video says), the problem is that there is simply too many of them. They are dominated by solvable groups, in fact most groups are just extensions of one abelian group by another, and the factor n^3 comes from cohomology. Roughly, cohomology is determined by choosing a representative on each generator pair, so if abelian group has p^n size, and is extended by a group of n generators, you get p^n^3 non isomorphic groups - homomorphisms are on the other hand determined by choosing value at each generator, there are much less homomorphisms. Computing cohomology can be challenging, but you can think of it as terms in uppper right corner of group representation by block matrices. You can basically choose that part arbitrarily when extending groups (provided that when you go to 1, your left upper corner goes to 0 - multiplication of matrices essentially acts like addition on upper right corner of block diagonal 2 by 2 block upper triangular matrices (for trivial action of G on Z(K) it is just addition, conjugation is represented with I matrix with addition of that upper right block matrix, when you multiply two of them, you get sum) so cohomology depends on the abelisation of G, which tracks how freely you can choose these upper right corner matrices, to ensure that map is homomorphism), with some group actionwhich is why cohomology depends on group action. Say you have a matrix representation of group K, and a matrix representation of group G, then you look at the matrix representation of K extended by G. If it were a direct product, you would have bock diagonal matrices, semidirect product adds action on the group by conjugation; on center, however, you have this action represented not only by diagonal matrices (which are I on G, and some matrix on center, which you can use as a linear space/ring where your group acts by conjugation), but you also have these right corner blocks which can basically be chosen arbitrarily, and correspond to cohomology - that is one way to look at these extensions). So yes, groups can be, based on prime groups and what is known about extensions basically 80 years ago and even in the 19th century (socles and other characteristic subgroups), systematically be listed, but sheer number of groups - mostly those which are solvable, i.e. have all factors just cyclic groups - is the obstacle, not lack of systematic way to approach. Indeed p groups have been counted explicitly for groups of order up to p^7, and there is algorithm how to do this in general and formulas how many of the groups there are - but it gets more complicated the larger groups one considera, though it is quite clear what to do. Basically these extensions are so numerous and complicated, the whole theory of group representations is contained in this classification of all finite groups - it is not lack of systematic approach, but the sheer richness of structures that appear that makes this a complicated thing, and mostly it is about these solvable groups. Because the number of non isomorphic group classes is so huge, only groups of sizes of up to few thousand elements have all been listed, and there are trillions of them.
31:50 Ah yes, the root of all evil...
sqrt(a)*sqrt(b) ~= sqrt(ab) :(
So does this show what group the galaxies that are 30 billion light years away are in?
Not with that mindset
2:32 😂
Awesome awesome video!