Introduction to Group Theory

[SuperHempo]

very nice :)

[jmbs0088]

coooooool ,man Im so high!!!! i got it all!!! im a step closer to M theory dude!

[Yaldatchayil]

Thank you! I'm going to raid your channel. I'm a music student studying transformation theory (an application of set/group theory) and I want to beef up my math background.

[yatdal]

Everyday you're shufflin' !! :D

[neogeo328]

Can you teach us more group theory please? :D

[theoya]

@neogeo328 +1

[LewisAM37]

i love your blue background

[leemanjoo]

I saw you at the preview of numberphile!

[singingbanana]

@leemanjoo Awesome!

[leemanjoo]

@singingbanana I have a question, I am interested in group theory that analyzes the crystal structure (space group, etc). What are some group theories that are involved when analyzing crystal structure?

[singingbanana]

@leemanjoo Crystallographic Groups.

[lekoman]

What I would find most helpful is a discussion of how group theory is useful. I don't doubt for a moment that it is, but as a complete layman, it seems to me as if this is stuff that's sort of easily taken for granted. Can you talk about some of the practical applications of this?

[DaTux91]

@lekoman The study of all solids has a lot to do with these kinds of groups. Geometrical symmetry is present in almost every crystal and even general solid. For example, the conductivity and optical diffraction properties of any material are closely related to its internal structure and symmetry. If we know that structure, we can predict these properties and take advantage of them. Semiconductivity and superfluidity are two examples of this. Piezoelectricity is another. All very interesting :)

[lekoman]

@DaTux91 Thank you! I love the part of YouTube where the thoughtful people hang out. ;)

[DaTux91]

@lekoman You're very welcome :) This channel is one of those places, and I love that too ;)

[singingbanana]

@dylanparker72 Groups act by permuting elements of a set, or by being written as a matrix and acting on a vector space. Symmetric groups permute the numbers 1 to n - like the cards in the video. Dihedral groups permute the corners of polygons - like the square in the video. The General Linear Group is the group of matrices and acts on vectors. So does the Special Linear Group, preserving volume, and the Orthogonal Groups, preserving length and angle.

[singingbanana]

@dylanparker72 Like I say, 'symmetry' in the mathematical sense means 'invariant'. Groups act on something, but there is some defining property that I wish to preserve. Even if it's just the size of the set.

[doudy009]

everyday he's shuffelin'

[jangofett333]

Can you tell me which math courses do I need study in order to take group theory?

[prAna530]

@jangofett333 -- "Can you tell me which math courses do I need study in order to take group theory?"

You will need to complete Linear Algebra and its prerequisites. It is then recommended to have some exposure to proofs. Universities usually have an "Intro to proofs" course that will help you develop the necessary skills for upper-division mathematics.

[KingofGames012]

this is nothing like the group or ring theory i study here. i thought for a set to satisfy as a group it need to satisfy few properties namely closure, associative, identity, inverse and abelian. now im really confused.

[singingbanana]

@KingofGames012 We are talking about the same thing.

[alquiora]

@KingofGames012 rofl

[alquiora]

@KingofGames012 you're not that good in recognizing patterns are you?

[KingofGames012]

@alquiora what patterns xD.

[alquiora]

@KingofGames012 Isn't it obvious from the very beginning when you learn isomorphism (which is one of the earliest topics in abstract algebra) that it implies some sort of symmetry? Of course there are a lot more about symmetries.

[KingofGames012]

@alquiora im almost aware of all the concepts like kernel, homomorphism, automorphism, isomorphism etc... all they told me is that isomorphism is a one- one, onto mapping from a group G to H but never told me where it is used or anything lol. im really getting sick of remembering 10s of theorems at once without knowing what are its applications where these things are used . =_=

[bluepie88]

@KingofGames012 Get a new textbook and/or teacher. Group theory has tons of geometric interpretation, and you're missing out if you're not being taught it - not least because you probably think it's far harder and more abstract than it is. :)

[KingofGames012]

@bluepie88 cant be helped dude. in my country all they teach us is theorems and theorems and theorems again instead of teaching applications and other stuff. i failed 2 times already in group / ring / analasys lol. wish i can pass this time. :(

[pspicer777]

@KingofGames012 What you describe is the very foundation of group theory. Once you start to add other structures and operations (invertible matrices, matrix multiplication) you get to General Linear Groups (for example). Add other structures such as orthogonal matrices and apply the resulting group over a field (Orthogonal Group) then you have a sufficiently rich structure to create Rotation groups (SO3) - the starting point for the exposition by singingbanana.

[LokiClock]

@KingofGames012 Groups don't need to be abelian, and the rotation group isn't.

[nirvana4ol]

Awesome pal... Thanks a lot... :)

[kinghezz]

this confuses meh

[katiekawaii]

You are super cool. I love living in a time where I can go on youtube and have brilliant people explain things to me that I would once have had to devote my entire education to in order to hear. The internet is awesome, 2011 is awesome, this guy is awesome.

[singingbanana]

@katiekawaii You're awesome.

[leaveblank]

@katiekawaii totally agree with you, I think you've spoken for many

[CCace61]

Thank you for this.

[TheDarkMessenger01]

Great examples, thanks :)

[spudw2k]

Thank you for sharing.

[coffeeiv]

Oh my gosh; I can't listen to him because he's got such beautiful hands! I'm thinking he could have been a pianist, cellist, or guitarist. However, he is a pretty interesting teacher, especially as young is he looks. He looks about seventeen.

[FHomeBrew]

I'm following classes into this subject but it's not that hard to understand yet :p Which I paradoxically find disappointing since I like a challenge. Good video :-) This is why I subbed :)

[orcodrilo]

I didn't know the dihedral group of four elements was coxeter! 

Thank you!

[FallofDarkness55]

i'm on my journey to understanding string theory mathematically.

[EclecticSceptic]

Yeah groups is mad, it's one of the option topics on my course (final year of secondary school), but the teacher isn't doing it - probably because most people's brains would explode - it's so cool though, nothing like the calculus or trig or binomial or whatever else we usually do.

[pagham1]

Thanks.

[gorgolyt]

cock WHAT groups!?

oh.

[UberSecret]

You never did the redundant dysfunctional anachronism of using a chalk board to slowly write down what you were saying while you were talking; and you used ready made 3 dimensional objects preplanned for immediate display--you deserve an Infinity of thumbs up, just for that alone!

[makier01]

Why only one lecture? :( Very interesting...

[elcanaldegeek]

If you could have subtitles in spanish :(

I will sub you anyways :D

[luislinuxero]

i'm a math student & i'm glad there's people like you spreading math in such an interesting way :)

[comecra85]

So, how this knowledge is used in solving tasks?

[thefetishkingfarts]

@comecra85 ah the pragmatic engineer!!!

[KevBaz78]

The length of time of this video is a cyclic permutation of the digits of Pi to two decimal places.

[buddhika216]

Inorganic chemistry sucks

[doggylikesoup]

Wow! Thank you so much! I am currently taking algebra and it is sooo hard to visualize the groups! Thank you for making this!!

[singingbanana]

@ZEALWANES A cube has 8 corners, so that's 4 pairs.

[ZEALWANES]

@singingbanana Ah, yes. Corners.

[ZEALWANES]

A cube has 4 pairs of opposing corners?  6 sides = 3 pairs, no?

[shikamarusgirl17]

This is really interesting. XD

[TheMedKing]

U rule munnnnnnnnnnnn. but bro can u tell me, as Im in high school, where can i find the necessary materials for a beginner.

[singingbanana]

@TheMedKing I would recommend 'popular science' books, that will tell you all the cool stuff and show you some technical stuff without losing you. You will have time to learn the technical stuff at university. To begin with I recommend books by Simon Singh, should be cheap on amazon, then explore the amazon recommendations.

[mgoerner]

@TheMedKing I recently was pointed out to a series of books called "A Mathematical Gift" (Uena/Shiga/Morita). It explains stuff from the ground up, with lots of pictures and examples and analogies in every day language. But it is more topology/geometry than group theory/algebra. But if you liked the way singingbanana introduced groups through symmetries, then this might be well suited for you.

I also really loved the more advanced "What is Mathematics?" (Courant/Robbins).

[DonCorleoneQ8]

I learn in 3 min more than I read in 3 chapters !

[Foaman]

Thank you. Tell me more. Great. Awesomely spoken. Well done.  Incredible. Fantasticating. Fascitastic.

[serpentblood1]

you make math interesting. and i hate math.... :)

[Tpal91]

Very interessting video

very well explained

Would you mind if i use one of your examples in school?

[singingbanana]

@Tpal91 Not at all.

[hanyahamba]

thanks.this is cool.

[Kewickviper]

I can remember doing this last year. I have to say you make it seem 100x more interesting than the maths lecturer I had. He just wrote proposition after proposition and never really explained anything.

[singingbanana]

Aww. I try to include bits like this in my lectures, as well as proposition info-dumping.

[jazzcazjazz]

nice

[sgtforge21]

james, why is this your only vid that has 1080p?

[11Agamemnon235]

Sounds like a lot of fun. Is a lot of this covered in undergraduate abstract algebra?

[mindissilent]

Yes, abstract algebra covers group theory.

[theboombody]

I'm glad I found this channel. I stopped my mathematical study at the undergraduate level, when I took real analysis and found out how calculus worked. For some reason I avoided the group theory classes.

I do have some interest in cardinality though, and its paradoxical implications.  I've argued with VeritySeeker before, and he's a good mathematical youtube source as you mention in another comment. You're a bit more user friendly though.

[69erthx1138]

Very clear introductory instruction. For a mathematician your quite good at turning formalism into common sense.

[singingbanana]

Thank you, (I think) :)

[giovanni9107]

@69erthx

Someone who focuses on formalism rather than reason when explaining basic mathematics is just a troll. There are plenty of trolls everywhere and it's because of them that most people gaze so much aversion to math. By the way this comment wasn't meant to sound like some sort of agression , it's just i had to get it off my chest. =)

[69erthx1138]

Agreed completely, and an insecure troll at that! (the whole, I'm smarter than everyone bullshit). This is why I chose to study physics because there is common sense and logic behind all the math fluff that one must prod through to make it work. Know many inflated math types personally, gotta take a pin to 'em every once a while :-{}

[singingbanana]

Yay for you guys :)

[mrandie80]

Really cool stuff, guy, but i'd like to see numerical examples along with this!!

[grayxy]

Nice. Keep a good work.

[TomRiverstone]

if you did a whole series on abstract algebra, especially the second part of the course, I'd say you were a YouTube hero

[singingbanana]

Find VeritySeeker on YouTube, he does it.

[TomRiverstone]

thanks

[translation314]

great job! you communicated very clearly

[acmartinezq]

Me parece una muy buena presentación, hace que desee conocer mas acerca de teoria de grupos

[Theowest]

Great teacher! :D

[111megamagic]

wow your smart

[zyzzyzus]

I am learning about group theory at the moment and this is just an introduction. What I am learning is more advanced than this but still is only the basics of group theory.

[ap2012]

You're so nice and smart! I hope by watching your video, some of your brainpower rubs off on me.

=D

[reclusivesage]

I wish I could understand this stuff. It would be cool to understand why the big, unsolved problems in math are still unsolved. I started to get lost around the 1:30 mark.

[jimpa93]

<3 ya singingbanana

[nadinelove27]

i wish you were my math teacher :) its crazy how you come up with so examples and are so into math

[JimmerSD]

More please.

[ortega24024]

Great: A reminder to learn abstract algebra (tried chatting up local uni's new lecturer on it to get her to help me out). Never settled into learning it cos I found the formulae small and boring. Also was disillusioned that when1st wanted to learn it, it was to understand characters of finite abelian groups, bought Allenby's book (my lecturer Wiegold was his tutor once) but found no mention of characters. Nick worked out the isomorphism between A_5 and rotations of the icosahedron himself.

[ALANxProductions]

I bet your old tutors and professors must be really proud of you.

I personally dislike maths, but after every video you make it more and more likeable. Even the advanced things you talk about I seem to grasp pretty easily.

Keep up the great work.

[zzScopoProductionszz]

i love this guy as much as i love math! keep making videos to keep me alive :D

[emjay1264jr]

this branch of math seems like it has a very large array of things

[Complexxx123]

i dont understand how can you study fliping shuffling and turning and what not. that area seems kinda short ended to study

[PJustinD]

No he simply explained an introductory of what he studies.. Im not very sure either, but I know its more complicted than simply flipping pieces of paper.

[uneedhelp2]

This is an introduction.. I'm sure what he studies is WAAYYY more complex and interesting.

[rotnakleugim]

group theory can apply to a rubik's cube, rihgt?

[singingbanana]

Exactly, like my examples you want to rotate the pieces until you get back to where you started - the solved position. The maths for the Rubik groups is quite advance and might not be accessible in a short video, but I might try in the future.

[dragonrid204]

honestly i am so entertained by your video's tbh i'm in 10th grade and even i understand what you speak about you explain it so well! i mean their is 1 or 2 things here and their that i may not understand but the rest of it is like ''zomfg...i understand!!!!'' haha i wish my teachers were like this :(

[Reed12223]

haha me too! my calculus teacher is terrible! i wish i had him as a teacher!!!

[10257danny]

YEs me too. (what math are you taking? I am average so i take algebra two in 10th grade)i am a sophmore.10th grader

[Kaiserks3338]

you are truly my mentor !!!

[dave597]

thx, this was interesting. :)

[Wewuvwabbitz]

Do you have a Green Screen?

[singingbanana]

I wish I did. This was just filmed in a blue room.

[Wewuvwabbitz]

Oh ok. :P

[loukas1997]

i cant understand what you say at the end

[v0lv0xx]

great!

and cool idea with those numbers on the squares back^^

[singingbanana]

Glad someone noticed :)

[TyYann]

Me too

[Error081688]

haha I saw that too.

[ssankoo]

Nice video, I really like that you tell us what you do in an understandable way. Thanks!

[Strijdparel]

I can understand that you like it. You are so pasioned about your proffession.

[kid29a]

nice background, looks pro this way :] thanks for sharing... keep it up

[zh1412]

i wish you were my lecturer:(((

[alabangma]

nice i learned many things from you

[ericsurf6]

I want to be in your class!

[Wupdog]

Cool!

[aivvaronis]

I wish you would come to my scool for at least a day :D

[FlippyCatFanMan]

W00T

1st Rating

1st Comment

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Nice !!

I really learned Something :D

Keep up the good work :D