Yeah, it really isn't. Symmetry groups are just part of discrete group theory. Also, the distinction between symmetric groups and symmetry groups is kind of glossed over here, they're very different things.
I'm a first year math student and I like to look up my professor's field of study, but most of the time it's hard to understand what it is that they're studying. Like I hear algebra or topology and I think I understand what it means but when I look it up I end up more confused. I think it would be awesome if professors (at least the ones teaching first and maybe second year courses) would do a video or presentation in layman's terms about what it is that their specialization is in. Sort of like this video!
I just burst out laughing at almost 2 in the morning. Thanks. XD Edit: Only downside is that now I can't unsee it. I will forever see James as half-size. Until I watch a video where there's another person in the frame for scale, of course. >.>
As a solid state chemist I come across symmetry and space groups all the time. It's a very important part of developing new materials. Thanks for all the mathematicians who have worked on the subject and thus made my work a it easier. :)
Very interesting, I had heard of group theory before but never knew what it was about. thanks for taking time to give a bird's eye view about the subject. I'd love to see ones on other types of math like this.
I've been looking for this kind od video for a long time. Simple and clear Introduction to these higher mathematics. Just so I can have a basic idea about this kind of math. Thank you very much sir. I'm reaLly hoping you would do this more often.
@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.
+SuperYtc1 Well, you can definitely prove Fermat's little theorem by a simple counting argument, although I don't know if that proof generalizes. Do you know of a proof of the more general Euler's totient theorem that doesn't involve group theory or by taking the phi(n) residues relatively prime to n?
not only that, but as you get to groups with more structure on them and when viewed homologically, there are entire classes of groups that are homomorphic, or even different entirely with the same homological properties (preserving 'exactness', preserving certain properties like 'solvability') not only on these classes of groups, but on describing all the possible maps between such groups (which is often another group, or a ring).
these are pretty advanced inorganic chemistry theorys, but if you're interested, here's a few: we use symmetry to simplify Schrodinger equations, so we don't have to do calculations for the whole molecule. other examples include -ligand to metal charge transfer theory, -spectroscopic theories -coordination chemistry -nomenclature of molecules and many others
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.
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.
This was a really cool video, thanks! I've just finished my third year at Manchester and I did a project last term on representation theory where I computed the character table of the B3 Coxeter Group, great fun!
You are everything that's right with this world, it inspires faith in humanity. I love knowing that there actally are people out there that find maths and such to be true beauty, and not the jersey shore.
I would love to see more of this "introduction" videos! you already did one or two about topology I think but not with that title. I wish you did more so people (like me) could have an idea of just how diverse Mathematics are.
@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).
@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.
@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.
Also you would want a book on Lie Groups. Those are the important physics groups (SU(n), SO(n) are expecially important in particle physics and quantum mechanics).
@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 :)
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.
I'm not sure what you are calling pompous but in case you are calling his PhD in combinatorial representation theory of the symmetric group pompous, you should know that is an area of mathematics.... it's not pompous, it's literally just the title...
it would be great to see some applications of group theory...and maybe a bit of the history to give it context...perhaps why it came about historically. like many math subjects, just explaining what it is seems limited.
@@Drkwll Yes. Yang-Mills devised the Standard Model of Quantum Mechanics which unified Electro-Magnetism with the Weak and Strong Forces: U(1) x SU(2) x SU(3) These are all groups. U(1) is for Electro-Magnetism. SU(2) is for the Weak force. SU(3) is for the Strong force. Georgi-Glashow then elaborated on this with their recent Grand Unified Theory, extending it to: U(1) x SU(2) x SU(3) x SU(5) this predicts proton decay which the Super Kamiokande experiment running under a mountain in Japan may confirm takes place. There is a pattern here which is mathematically defined: 1 + 2 = 3 2 + 3 = 5 This suggests a further unification may be possible, by continuing this sequence: 3 + 5 = 8 This would then look like: U(1) x SU(2) x SU(3) x SU(5) x SU(8) Georgi-Glashow elaborated their SU(5) theory to a SO(10) or Spin(10) theory, by taking the "double cover" of SU(5), only for it to then predict a lot more particles that haven't been found by the Large Hadron Collider. It is possible that SU(8) being more restrictive wouldn't suffer from this problem and could unify everything else, including all the idiosyncracies of the broken symmetries arising from CPT - i.e. Charge, Parity, and Time. It is strange that we only see unequal amounts of matter and anti-matter, only some of the right handed versions of particles, and no evidence of there being an anti-time that flows in the opposite direction (even though Richard Feynman said that a positively charged position could be regarded as a negatively charged electron that was going backwards in time). Unfortunately, it may be impossibly expensive, or dangerous to build the supercolliders necessary to reach the unification energies to even reveal a set of right handed versions of existing particles which were hiding in an imaginary mirror universe, let alone recreate the circumstances of the Big Bang without making a new one that could replace our universe.
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?
Simple groups are basically the prime numbers of group theory. All groups are either simple, or formed by combining two or more simple groups together.
Do not know but equivalence classes is important in set teory so it is probably important in everything, and that is probably why they teach it in beginners math courses. The natural numbers if counted modulus like witch numbers divided by 7 has the same remainder seams like a equivalence class when it divide the natural numbers in distinct subgroups.
You explained very well using square and triangle. My teachers taught theoretically.... It was so boring n I am still confused.....can you make more such videos taking ex of many molecules....
@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.
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 :(
Is this the same thing as point groups in mineralogy? For example a cube is in the point group 4/m bar3 2/m. Meaning it has a 4 fold 90 degree rotational axis through the center of an arbitrary one of its faces and perpendicular to a mirror plane. The vertex of three faces is a 3 fold 120 degree rotational axis perpendicular to an inverted mirror. And the center point of the intersection of 2 sides is a 2 fold 180 degree rotational axis of symmetry perpendicular to a mirror plane.
OK I have always wondered why it seems to be a good thing to establish that some binary operator on a closed set has a group structure. This is probably the best answer, considering that there is a limited number of isomorphic group structures especially when you count out the cyclic ones. Now I just wonder why a simple group is good and how common they are, there is only 23 ore so isomorphic ones so they seems important.
oh, so the more lay term is just finding the "function family" and then knowing how each change to the inputs affects the graph as a whole? I remember doing that a lot in calculus :D
Way is there so little lectures on "Mathematical logic" out on the internet. Most logic is very basic and do not explain for example completeness and compactness in predicatelogic or functions in different semantics. You are a math pro and very good, please do same! You do not have to do a complete series when basic stuff is already out there like truthtabels and so but the mathematical approach is missing the hard thing is understanding this union thoughts.
Prof Hugo de Garis has done one lecture series in Group theory and he is following a book, that I bought on internet for like 20 dollar used. Not so many people seems to care to take in what he teaches. Only like 20 views probably because his technical filming and presentation is a bit primitive, but all right if you put the sound low and max out the writhing. I had all ready finished my course in "abstract algebra" when I follow Prof Hugo De Garises course but I think I kind off got it then.
@John Smith This video doesn't say that much about what group theory is useful for, which is why it may not seem that interesting. However it is really foundational to most of mathematics and modern physics. If you want a career in math you will run into it in some form or another anyway. It is not strictly necessary to take a separate course in it, but it is useful to do it just to get used to the concepts and way of thinking.
I stand corrected they seem to be a finite classification, but of infinite sets of groups: "The list of finite simple groups consists of 18 countably infinite families, plus 26 sporadic groups that do not follow such a systematic pattern." (Wiki)
You can find it under "Abstract Algebra" although not all abstract algebra is group theory. Rings and Fields are also a part of it. Google Klein four group and Galois. to get you started. Also a book called Algebra by Michael Artin (Google him as well)
I keep getting distracted by trying to assess is the red markings for the numbering bled through, or if James just perfectly wrote them backwards on the flipped side.
Any intro book would do you good. Papantanopoulu or Dummit and Foote. You would probably be more interested in the study of group actions (which are what make groups important in physics) so if their is a book that focuses on group actions, that might be good for you, but I don't know of a book like that sorry.
Hi ! I´m a physics student and I´ve been asked by a teacher to study group theory on my own. Could you recommend me a group theory book for begginers? And now that I´m writting I´d like take the opportunity to tell you that I think you do an awsome job with the videos, I´m kind of a fan of yours actualy, I´ve seen you here and in Numberphile (you are my favorite talker from numberphile) and you always give excelent explanations, keep up the good work James
"A Book of Abstract Algebra" by C.Pinter is an excellent book for self-study. It will gently introduce you to group theory with a very natural language and strong motivations. Good luck!
James, if you by chance see this comment, thanks for all the great videos. I'm a maths undergraduate student and have been looking for a good book on group theory. Is there any you would recommend?
Hi James, I was wondering whether you could make a video exploring the group theoretic aspects and symmetries of the solutions to a 3-d game of tic tac toe? thanks in advance
@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?
@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 . =_=
He is reading. Now, that I see it, I can't 'unsee' it anymore. I like it he speaks to me rather than when he reads to me. Although I have to say he is a good actor. PS- I love abstract algebra
singingbanana you really replied?! Thanks. And I could see that you were not looking into the camera. That's why I thought you were reading out. BTW, I did not mean to offend. I really like your videos.
I wonder if there is a relationship between the 5 platonic solids and/of they can fit in the 17 "wallpaper" groups. Let supose they do, can these 17 "wallpaper" groups found in higher symmetrie groups until 65536 + 1 , ... ? So this suggest there is somehow a relation between twinprimes and symmetrie groups? (just by intuition)
I need your help understanding something. I once thought it may be possible that anything divided by zero would get you not the quantity zero, but the absence of a quantity that zero can substitute for. It's a long explanation that I don't plan on getting in to, but bear with me. I posted this on a science forum, expecting my idea to get shot down by a long, complicated branch of mathematical facts that I knew nothing about. Surely enough it did. A person told me that division by zero getting the absence of a quantity would break the rules of group theory. He was talking about something like this, right? 10/2=5 10/5=2 Because, if so, then I think I may have an idea on how to fix my idea. It's a long shot, but it may just work.
Yes I've seen these kinds of descriptions many times. My question however is: but why does group theory describe particle physics? As best I know it originated with Wigner and if I better understood the work he did, I'd be better able to answer my question - but I'm not there yet.
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.
"Group theory is the study of symmetry."
I certainly wasn't expecting _that_. 12 seconds in and my mind is already blown.
Yeah, it really isn't. Symmetry groups are just part of discrete group theory. Also, the distinction between symmetric groups and symmetry groups is kind of glossed over here, they're very different things.
I'm a first year math student and I like to look up my professor's field of study, but most of the time it's hard to understand what it is that they're studying. Like I hear algebra or topology and I think I understand what it means but when I look it up I end up more confused. I think it would be awesome if professors (at least the ones teaching first and maybe second year courses) would do a video or presentation in layman's terms about what it is that their specialization is in. Sort of like this video!
2:13 I never knew James was such a small man.
I just burst out laughing at almost 2 in the morning. Thanks. XD
Edit: Only downside is that now I can't unsee it. I will forever see James as half-size. Until I watch a video where there's another person in the frame for scale, of course. >.>
haha, i think those are just big cards!
lol
Bob Bobson what of he is normal and we are all giants?
wow lol
Can you do an entire series on this? I would watch it and give you my first child.
Orpheus a chemistry Grad student who likes physical inorganic chemistry agrees
Orpheus also might I add one that loves group theory .... but isn't very good at it
wtf
As a solid state chemist I come across symmetry and space groups all the time. It's a very important part of developing new materials. Thanks for all the mathematicians who have worked on the subject and thus made my work a it easier. :)
pleese do more group theory video
Very interesting, I had heard of group theory before but never knew what it was about. thanks for taking time to give a bird's eye view about the subject. I'd love to see ones on other types of math like this.
Yes! Totally cool, and important.
Thank you for sharing your interest. Your explanation was very helpful. I liked your insight into shuffling and the potential applications therein.
I've been looking for this kind od video for a long time. Simple and clear Introduction to these higher mathematics. Just so I can have a basic idea about this kind of math. Thank you very much sir. I'm reaLly hoping you would do this more often.
@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.
Oh my god, group theory sounds SO interesting. Abstract algebra seems really cool, from my brief research.
It's pretty neat. Fermat's little theorem is easily proved using basic group theory.
+Richard Spence Fermat's theorem is actually proved by using primary school math.
+SuperYtc1 Well, you can definitely prove Fermat's little theorem by a simple counting argument, although I don't know if that proof generalizes. Do you know of a proof of the more general Euler's totient theorem that doesn't involve group theory or by taking the phi(n) residues relatively prime to n?
I always thought what I learned in abstract algebra was totally useless. You explained it better in a couple of minutes. Thanks, dude.
not only that, but as you get to groups with more structure on them and when viewed homologically, there are entire classes of groups that are homomorphic, or even different entirely with the same homological properties (preserving 'exactness', preserving certain properties like 'solvability') not only on these classes of groups, but on describing all the possible maps between such groups (which is often another group, or a ring).
Very clear introductory instruction. For a mathematician your quite good at turning formalism into common sense.
Keep sharing your knowledge and enthusiasm for math. You've inspired me to continue my studies.
these are pretty advanced inorganic chemistry theorys, but if you're interested, here's a few:
we use symmetry to simplify Schrodinger equations, so we don't have to do calculations for the whole molecule. other examples include
-ligand to metal charge transfer theory,
-spectroscopic theories
-coordination chemistry
-nomenclature of molecules
and many others
Nice video, I really like that you tell us what you do in an understandable way. Thanks!
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.
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.
This was a really cool video, thanks! I've just finished my third year at Manchester and I did a project last term on representation theory where I computed the character table of the B3 Coxeter Group, great fun!
You are everything that's right with this world, it inspires faith in humanity. I love knowing that there actally are people out there that find maths and such to be true beauty, and not the jersey shore.
Very clear introduction to group theory. Congratulations
U really study crazy things.
I love it 👍
I would love to see more of this "introduction" videos! you already did one or two about topology I think but not with that title. I wish you did more so people (like me) could have an idea of just how diverse Mathematics are.
@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).
@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.
@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.
@ZEALWANES A cube has 8 corners, so that's 4 pairs.
great! I am in the middle of a course on Scientific Reasoning! Needed this video to get me through the readings!
Ok, I understand now way it is important.
It seems quite amazing that it would only bee like 23 primes for groups.
Cool explanation!!!
I want to be in your class!
Count me in!
Absolutely, resolutely and relavantly fantastic!!!!!
Thank you. Tell me more. Great. Awesomely spoken. Well done. Incredible. Fantasticating. Fascitastic.
i love this guy as much as i love math! keep making videos to keep me alive :D
Interesting introduction. Thank you.
damm i can watch this guy for 10000000 hours, man please upload longer group theory vids u r awesome!!!!
I watched the live numberphile video earlier today. So I thought I'd learn about group theory an you were the first vid that popped up.
Also you would want a book on Lie Groups. Those are the important physics groups (SU(n), SO(n) are expecially important in particle physics and quantum mechanics).
Wow! Thank you so much! I am currently taking algebra and it is sooo hard to visualize the groups! Thank you for making this!!
@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 :)
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.
I didn't know you were a group theorist. I'm curious what subarea. Do quasigroup theory next (I'm kidding)
My PhD was combinatorial representation theory of the symmetric group.
I'm not sure what you are calling pompous but in case you are calling his PhD in combinatorial representation theory of the symmetric group pompous, you should know that is an area of mathematics.... it's not pompous, it's literally just the title...
singingbanana What is CR?
singingbanana where may I read your PhD, a link would be well appreciated
it would be great to see some applications of group theory...and maybe a bit of the history to give it context...perhaps why it came about historically. like many math subjects, just explaining what it is seems limited.
There's huge applications of group theory in physics.
@@Drkwll Yes.
Yang-Mills devised the Standard Model of Quantum Mechanics which unified Electro-Magnetism with the Weak and Strong Forces:
U(1) x SU(2) x SU(3)
These are all groups. U(1) is for Electro-Magnetism. SU(2) is for the Weak force. SU(3) is for the Strong force.
Georgi-Glashow then elaborated on this with their recent Grand Unified Theory, extending it to:
U(1) x SU(2) x SU(3) x SU(5)
this predicts proton decay which the Super Kamiokande experiment running under a mountain in Japan may confirm takes place.
There is a pattern here which is mathematically defined:
1 + 2 = 3
2 + 3 = 5
This suggests a further unification may be possible, by continuing this sequence:
3 + 5 = 8
This would then look like:
U(1) x SU(2) x SU(3) x SU(5) x SU(8)
Georgi-Glashow elaborated their SU(5) theory to a SO(10) or Spin(10) theory, by taking the "double cover" of SU(5), only for it to then predict a lot more particles that haven't been found by the Large Hadron Collider. It is possible that SU(8) being more restrictive wouldn't suffer from this problem and could unify everything else, including all the idiosyncracies of the broken symmetries arising from CPT - i.e. Charge, Parity, and Time. It is strange that we only see unequal amounts of matter and anti-matter, only some of the right handed versions of particles, and no evidence of there being an anti-time that flows in the opposite direction (even though Richard Feynman said that a positively charged position could be regarded as a negatively charged electron that was going backwards in time). Unfortunately, it may be impossibly expensive, or dangerous to build the supercolliders necessary to reach the unification energies to even reveal a set of right handed versions of existing particles which were hiding in an imaginary mirror universe, let alone recreate the circumstances of the Big Bang without making a new one that could replace our universe.
I can understand that you like it. You are so pasioned about your proffession.
So interesting, I wish you went into more detail...
Make it into a series!!
Good energy and well explained.
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?
Simple groups are basically the prime numbers of group theory. All groups are either simple, or formed by combining two or more simple groups together.
Do not know but equivalence classes is important in set teory so it is probably important in everything, and that is probably why they teach it in beginners math courses. The natural numbers if counted modulus like witch numbers divided by 7 has the same remainder seams like a equivalence class when it divide the natural numbers in distinct subgroups.
Very interesting topic. Any video recommendations to learn about this?
You explained very well using square and triangle.
My teachers taught theoretically.... It was so boring n I am still confused.....can you make more such videos taking ex of many molecules....
@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.
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 :(
Is this the same thing as point groups in mineralogy? For example a cube is in the point group 4/m bar3 2/m. Meaning it has a 4 fold 90 degree rotational axis through the center of an arbitrary one of its faces and perpendicular to a mirror plane. The vertex of three faces is a 3 fold 120 degree rotational axis perpendicular to an inverted mirror. And the center point of the intersection of 2 sides is a 2 fold 180 degree rotational axis of symmetry perpendicular to a mirror plane.
OK I have always wondered why it seems to be a good thing to establish that some binary operator on a closed set has a group structure. This is probably the best answer, considering that there is a limited number of isomorphic group structures especially when you count out the cyclic ones. Now I just wonder why a simple group is good and how common they are, there is only 23 ore so isomorphic ones so they seems important.
that observation was awesome
oh, so the more lay term is just finding the "function family" and then knowing how each change to the inputs affects the graph as a whole? I remember doing that a lot in calculus :D
Way is there so little lectures on "Mathematical logic" out on the internet. Most logic is very basic and do not explain for example completeness and compactness in predicatelogic or functions in different semantics. You are a math pro and very good, please do same! You do not have to do a complete series when basic stuff is already out there like truthtabels and so but the mathematical approach is missing the hard thing is understanding this union thoughts.
so simple explaining idea of so complex subject.
Prof Hugo de Garis has done one lecture series in Group theory and he is following a book, that I bought on internet for like 20 dollar used. Not so many people seems to care to take in what he teaches. Only like 20 views probably because his technical filming and presentation is a bit primitive, but all right if you put the sound low and max out the writhing. I had all ready finished my course in "abstract algebra" when I follow Prof Hugo De Garises course but I think I kind off got it then.
Me parece una muy buena presentación, hace que desee conocer mas acerca de teoria de grupos
I didn't know the dihedral group of four elements was coxeter!
Thank you!
I love your work! Thank you :')
Nice. Keep a good work.
Good idea.
you are truly my mentor !!!
@John Smith This video doesn't say that much about what group theory is useful for, which is why it may not seem that interesting. However it is really foundational to most of mathematics and modern physics. If you want a career in math you will run into it in some form or another anyway. It is not strictly necessary to take a separate course in it, but it is useful to do it just to get used to the concepts and way of thinking.
Magnus Dahler Norling chemistry/quantum mechanics requires it
I stand corrected they seem to be a finite classification, but of infinite sets of groups:
"The list of finite simple groups consists of 18 countably infinite families, plus 26 sporadic groups that do not follow such a systematic pattern." (Wiki)
More group theory!
You can find it under "Abstract Algebra" although not all abstract algebra is group theory. Rings and Fields are also a part of it. Google Klein four group and Galois. to get you started. Also a book called Algebra by Michael Artin (Google him as well)
I keep getting distracted by trying to assess is the red markings for the numbering bled through, or if James just perfectly wrote them backwards on the flipped side.
Make it into a series!
Any intro book would do you good. Papantanopoulu or Dummit and Foote.
You would probably be more interested in the study of group actions (which are what make groups important in physics) so if their is a book that focuses on group actions, that might be good for you, but I don't know of a book like that sorry.
Hi !
I´m a physics student and I´ve been asked by a teacher to study group theory on my own.
Could you recommend me a group theory book for begginers?
And now that I´m writting I´d like take the opportunity to tell you that I think you do an awsome job with the videos, I´m kind of a fan of yours actualy, I´ve seen you here and in Numberphile (you are my favorite talker from numberphile) and you always give excelent explanations, keep up the good work James
"A Book of Abstract Algebra" by C.Pinter is an excellent book for self-study. It will gently introduce you to group theory with a very natural language and strong motivations.
Good luck!
Thank you very much :) Saad Slaoui
James, if you by chance see this comment, thanks for all the great videos. I'm a maths undergraduate student and have been looking for a good book on group theory. Is there any you would recommend?
Hi, I'm not sure if you will see this anytime soon but could you do a video on group theory for chemistry.
Hi James, I was wondering whether you could make a video exploring the group theoretic aspects and symmetries of the solutions to a 3-d game of tic tac toe?
thanks in advance
@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?
@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 . =_=
He is reading. Now, that I see it, I can't 'unsee' it anymore. I like it he speaks to me rather than when he reads to me. Although I have to say he is a good actor.
PS- I love abstract algebra
I don't think I was reading.
singingbanana you really replied?! Thanks.
And I could see that you were not looking into the camera. That's why I thought you were reading out. BTW, I did not mean to offend. I really like your videos.
@DaTux91 Thank you! I love the part of TH-cam where the thoughtful people hang out. ;)
@katiekawaii You're awesome.
@KingofGames012 We are talking about the same thing.
he's good :) can you have some more videos? like discussing about the point groups (low and high symmetry)
My fav teacher
I saw you at the preview of numberphile!
I LOVE this video! Thanks!
@leemanjoo Awesome!
@leemanjoo Crystallographic Groups.
I wonder if there is a relationship between the 5 platonic solids and/of they can fit in the 17 "wallpaper" groups. Let supose they do, can these 17 "wallpaper" groups found in higher symmetrie groups until 65536 + 1 , ... ?
So this suggest there is somehow a relation between twinprimes and symmetrie groups?
(just by intuition)
I need your help understanding something.
I once thought it may be possible that anything divided by zero would get you not the quantity zero, but the absence of a quantity that zero can substitute for. It's a long explanation that I don't plan on getting in to, but bear with me.
I posted this on a science forum, expecting my idea to get shot down by a long, complicated branch of mathematical facts that I knew nothing about. Surely enough it did. A person told me that division by zero getting the absence of a quantity would break the rules of group theory. He was talking about something like this, right?
10/2=5
10/5=2
Because, if so, then I think I may have an idea on how to fix my idea. It's a long shot, but it may just work.
That it would "break the rules" of group theory doesn't make sense. Zero is not a member of any multiplicative group.
Yes, it is.
Would you explain why is it possible to describe "the Prlblem A" as a group? Is it always possible?
Yes I've seen these kinds of descriptions many times. My question however is: but why does group theory describe particle physics? As best I know it originated with Wigner and if I better understood the work he did, I'd be better able to answer my question - but I'm not there yet.
Thank you for sharing.
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.
awesome video, thanks! You're amazing!
i dont understand how can you study fliping shuffling and turning and what not. that area seems kinda short ended to study
@Tpal91 Not at all.
Glad someone noticed :)