Dihedral Group (Abstract Algebra)

The Dihedral Group is a classic finite group from abstract algebra. It is a non abelian groups (non commutative), and it is the group of symmetries of a regular polygon. This group is easy to work with computationally, and provides a great example of one connection between groups and geometry.
Be sure to subscribe so you don't miss new lessons from Socratica:
bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/...
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : / socratica
► Make a one-time PayPal donation: www.paypal.me/...
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: / socraticastudios
Instagram: / socraticastudios
Twitter: / socratica
♦♦♦♦♦♦♦♦♦♦
Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
♦♦♦♦♦♦♦♦♦♦

[@Socratica]

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

[@helgemeyer8363]

"When you talk to your friends about group theory..." Yeah, like my friends cared about math. Excellent video, though.

[@Andre83572]

Friends? What are those?

[@brandonhughes645]

time to get new friends fool. lol

[@howmathematicianscreatemat9226]

those "friends" will soon not be able to be employed anymore when robots take over. Only creative jobs will survive. So you will surpass them later if they don´t respect your interests now!

[@Grassmpl]

You made the wrong friends

[@PMA_ReginaldBoscoG]

I don't know why but I read your comment in Electroboom Mehdi Sadaghdar's voice 😂

[@jessicalynn4415]

She really broke this down perfectly.

[@markdstump]

She doesn't write these, but her delivery is nice :)

[@judithbenami3964]

@@markdstump I am so upset I think she is AI...

[@markdstump]

@@judithbenami3964 Don't worry, Liliana​ ​de​ ​Castro is real.

[@jamesthomson4186]

"when you're talking to your friends about abstract algebra"...

[@tylerw.1414]

LOL THIS

[@rngwrldngnr]

Going through this playlist again trying to refresh myself to get back into Pinter's A Book of Abstract Algebra, hopefully to finish it this time. Its very educational and well made. What particularly stands out is how well the later videos were slotted into the playlist to cover gaps, it works better than I'd have thought possible.

[@oyaoya2468]

ngl you explained it really well. Its not that I was ungrateful to what my book has taught so far, but you has enlightened me a lot to digest such "abstract" concept.

[@RuthJatibOConnor]

Loved this. Thank you! As an artist, I’m constantly looking for visual representations of abstract themes and ideas. I became aware of the Dihedral group by a friend a few years ago and created an art piece illustrating this concept. So interesting!

[@dionsilverman4195]

Just fantastic presentation of this material. I wish I'd found this years ago. These are very clear, accessible, and engaging.

[@gracemisere6994]

And you are a wonderful presenter!

[@Socratica]

You are so kind! Thank you for watching. :)

[@MuhammadTahir-xi6ve]

I like your clear way of talking. I meant your lecture about dihedral groups is very clear and to the point. "May you remain healthy and happy "

[@franciscodanielquiroz9904]

You make me feel like I can understand every single topic in Mathematics. I love you!

[@tadashimori]

Not sure if it would ever be possible, but you should try to bring the polynomial function of degree 5 problem in an easy way for everyone to understand. Maybe solvable and super solvable groups... I remember studying like crazy to get these topics, and I'd love to see how you would explain it xD.

[@TheKiskus]

You've just earned yourself a new subscriber. This was a really great video that broke down the abstraction level to a point where it's understandable and now it's possible to understand the more complicated stuff regarding the subject. Thank you!

[@Socratica]

We're so glad you found our video helpful! This was wonderful to hear. We're so glad you are learning Abstract Algebra with us. Happy New Year! :)

[@TheKiskus]

I am studying math and taking a course about groups this semester and rings next semester. I am happy that I found your channel it's very easy to understand. Happy New Year to you as well! :)

[@saeedanwar5262]

dear how can i get relative theorems on each topic???????

[@xanadudeboston3344]

This is a perfect explanation of this topic. I came in asking, “what the hell is that?” and came out saying, “I’ll know what’s going on next class.” Great work. The animations spelt it out well.

[@celsogrodriguez1940]

Currently, I have a Bad Math Teacher and that is not because he doesn't know about the topics but He can not explain to us what is a dihedral group. Now with your explanation, I can understand what a Dihedral group means. I am very grateful to you. Best Regards from Guatemala.

[@Socratica]

We're so glad we could help. Try to find a good study group so you can help each other - hang in there!! 💜🦉

[@JohnCLiberte]

absolutely amazing :) love your humour. Who is doing these animations? They are of a very very high quality.

[@Socratica]

Thank you so much, Jacob! We're so glad you've subscribed to our channel! We're a small team here at Socratica. We do most of our work in the Adobe Suite (Premiere, After Effects). Thank you for your kind comment! :)

[@howmathematicianscreatemat9226]

yes, they are a didactical masterpiece!

[@joshjucutan]

So I'm working on a Senior Project with symmetries relating it to art, and you broke this down so beautifully!

[@Socratica]

Love this idea! 💜🦉

[@Socratica]

Socratica Friends! If you're wanting to improve your study skills, we WROTE THIS BOOK all about how we figured it out! How to Be a Great Student ebook: amzn.to/2Lh3XSP Paperback: amzn.to/3t5jeH3 or read for free when you sign up for Kindle Unlimited: amzn.to/3atr8TJ

[@RoyYakoobAlHakeem]

The last message was funny and made me subscribe!

[@SigmaSixSoftware]

This explained the whole section on dihedral sets better than my textbook thank you

[@jongxina3595]

Damn I wish you guys did more playlists like this on other math topics as well. You guys are awesome!

[@muhannadayoubi1433]

Her voice is very calm and her explanation is very clear and easy to follow. The last part cracked me up 😆 but I do have a question that has been lingering on my mind. When I studied Real Analysis, it was alway stressed that we must always use analytical methods and never rely on our geometric intuition when writing proofs (we can rely on it, of course, to get a feel of the problem and proof). But now in abstract algebra, we're freely discussing geometry and symmetries albeit I haven't seen anyone formalise what these symmetries mean. What do you think of this?

[@toasteduranium]

Let me know when someone figures this out

[@abouland9404]

Even the most abstract concepts in advanced math come from something more intuitive, whether it's examples, geometric intuition, or something else more concrete. It is always a good idea to be able to think of these examples when thinking of the definitions or theorems. When writing a proof, you do not use these ideas in your proof and instead keep it formal, which I think what was being emphasized during your Real Analysis class. Unfortuantely, many professors ignore the intuition and just give formal definitions and proofs without motivating them or giving a clear picture of the concept first. In this channel, the videos are meant for understanding, and not formalism, which is why they are clear and inutitive.

[@RolandHutchinson]

We could just say that a symmetry of a plane figure is a mapping from the figure onto itself that preserves distances and angles. (The preservation of distances implies that it is also 1-1, hence a bijection.) We could identify those mappings with linear transformations represented as matrices. Or, to keep things entirely analytical and algebraic, we can learn a bit more algebra define a symmetric group as the semidirect product of two cyclic groups (Z/nZ and Z/2Z, the latter intuitively representing the flip). But first we have to define and understand semidirect products. You can look this up if you are interested.

[@RolandHutchinson]

@@toasteduranium I made a couple of suggestions about how this might be done in my reply to @muhannadayoubi1433.

[@vivelesport8197]

Clearly done! Workmanly! For the reflection i suggest it to be done very slowly to see and understand how vertices are reversed 2 by 2... You deserve encouragement...!

[@sarkarimaster-mohanpatel6027]

Though I am not a mathematica student but I like to watch you. Your teaching is very attractive... I feel I need learn mathematics from very beginning...

[@HaiderAli-cp4rp]

A very good and comprehensive lecture on Dihedral groups... Please also upload videos on Alternating groups, transpositions, contingency permutations, even and odd permutations, signum of permutations, Simple groups

[@sanjursan]

Maybe show this vid BEFORE the vid on subgroups. Then you can demonstrate the two (non-trivial) subgroups of the equilateral triangle. Just a thought. Otherwise, the information and production of this series is EXCELLENT!

[@roscelledalman6814]

more videos about abstract algebra please

[@shwetabhsrivastava594]

Can you make one video related with group action and orbit? Thanks

[@Treviscoe]

Thanks for this video, you cleared up a confusion I had about the two styles of notation (D6 can either refer to the symmetry group of a hexagon or that of an equilateral triangle).

[@107ashrafulmasum5]

excellent job.you're amazing teacher

[@dzieseliezietsu3905]

Lol you got a new sub take this cookie 🍪 The ending got me smiling up at 4am in the morning. 😹😹

[@24chiku]

Please consider making a set of videos on point set Topology.

[@PunmasterSTP]

That was a really good heads-up about the notation; I will definitely di-heed your advice!

[@Monika_atri10]

I was searching for geometrical representation of Dn group and finally i got ur video ..its osm

[@dylano5595]

You should write a full abstract algebra textbook, written the way you speak now. That would be an awesome reference for me :-)

[@goodjunksam8149]

Your fluidity is as beautiful and abstract as yoga with Adriene. Higher math made fun. Next I think I will go over to watch Hans Rosling. Blessings.

[@prikarsartam]

some moments I doubt in what i'm falling in for, group theory or miss Liliana​ ​de​ ​Castro! Never mind please! Thanks a lot by the way for such wonderfully brief yet precise description.

[@mu.makbarzadeh2831]

Thank you so much! I really have become familiar with many things in mathematics by you!

[@wangyoutian]

Beautiful video. Thanks

[@ewabehr4861]

My group theory teacher is really shitty, this video was so helpful! Now that I know what it's even about I can go and learn formal definitions and theorems

[@nazeerjehan6550]

Just Amazing.. The way you teach is very lucid.👍

[@azhari7968]

The animation really helped

[@IjazKhan-fm4si]

I miss ur channel.ur skill communications are very amaizing

[@Grassmpl]

3:58 poor lady. Needs validation and all our kindness.

[@qaziarshad939]

More videos about Abstract Algebra, Real Analysis, and Topology please

[@drkevinsullivan]

A Rachel Madow of group theory! (That's meant to be a nice compliment, by the way.) Nice presentation. Thanks.

[@ritikanegi1604]

Thank you ma'am, just want to request you to make video on class equations.

[@ashfaqhussain6940]

Absolutely brilliant effort

[@howmathematicianscreatemat9226]

yes. It´s very sad she gets so little money.

[@laurentverweijen9195]

Can you make a video explaining conjugency classes and the charasteristic equation?

[@sandipagarwal0511]

Perfect, what I assume about dihedral group all are truly verified here. Perfectly done

[@santoshnegi5693]

Giving ideas is more important... That what u do.. Thanks💖

[@Bablu406611]

Please make a video in Galois theory

[@krishanbajiya4962]

Thank you mam your videos is very helpful my study

[@moularaoul643]

Thanks

[@ManpreetKaur-rm2wl]

so precise and good

[@nallawarvijay]

Excellent video.... I like your explain... Tq i wish make more videos... Do you have any website??

[@MrVictius]

Nice vids. Just earned yourself a subscriber

[@Xan_the_r]

Thank you so much. Your videos are amazing.

[@ben_jammin242]

Thank you. The subscribe button and I needed some privacy.

[@jonny1996ahh]

Here you go subscribed! I have an exam in abstract algebra in 2 weeks :D

[@kalpanachaudhary8211]

Thankyou so much...love your explanation :)

[@radicalpotato666]

Thank you very much. I have subscribed.

[@niyathisanand]

Thank you very much mam and good presentation 👍

[@ahmedsafiya6042]

clear and simple.

[@chandlerbing5434]

Great videos loved your explanation , please make more videos on group theory about the cyclic groups in details

[@kaneezfatima926]

Excellent video

[@gopalkhadka2077]

It's Awesome explanation by you ... Thanks for making this video😊

[@Rawat-en1rl]

Yeah sure ! Agreement on notation is a must while having a normal conversation Let me narrate how I lost a friend:- Me- " Hey dude how are you ? long time no see !" He- "hey I am fine, was busy with my abstract algebra assignment " Me- "oh! have you read about the D6 triangular group?" He - (Sighs disappointedly ) " I don't know this man" ( Walks away)

[@narendrakhadka9598]

excellent , i really understood

[@ratl0ver3]

I love this. You are wonderful

[@stuti-ruthjacob5246]

2n symmetries is due to all possible rotations and combination of rotation and flip about any 1 axis of the n sided polygon.. But it can flip about other axis too so there should be more than 2n symmetries... 🤔 Right?

[@Power321ify]

2:00 For n-gon with odd number of sides, it can be seen on your diagram that there are lines of symmetries which don't go through the vertices of n-gon, but you say that the line of symmetries of n-gon are the lines through the vertices of n-gon?

[@keepmoving5453]

you are a very good teacher mam

[@nazeerjehan6550]

Can you please make a video on the centre of dihedral group 🙏

[@MikeJones-ue7ux]

THANK YOU!!! Dummit and Foote is a great abstract book, but the section on dihedral groups is awful. I finally understand this.

[@derciferreira7211]

Very good !.You dont complicate it, Can you make a vedeo about group action please.

[@a.s2639]

Honestly i lost all hope in Group theory, cause of an horrible university teacher. But now that you expose things clearly, i mean, it's litteraly an other world! I can see again the light of comprehension and follow again the road of studies! So just wanted to say thank you.

[@Socratica]

This is so wonderful to hear. When we read a message like yours, it really inspires us to keep making videos. So thank YOU! And keep us posted about your studies! 💜🦉

[@jahirul9568]

Thank you very much

[@twilight1176]

Just beautiful

[@joshmart2569]

Add and solve difficult questions in your videos..... You will go far

[@shubhradwivedi7093]

.....what a nice explanation !!!!!!!

[@Socratica]

Thank you so much, Shubhra! We're so glad you are watching! :)

[@logantheyoyoman]

Awesome videos!!!

[@MrVictius]

Could you help me with figuring out how groups relate to solving polynomial equations? Specifically, can you explain Galois theory and what that has to do with the solutions of polynomial equations?

[@twilightsparkle6756]

MrVictius Polynomial equations are *closed* on the set of complex numbers, which means that any arithmetic operation used in a polynomial equation works inside the set and cannot escape it, therefore all solutions to such equation (its roots) are in this set. Now take a look at some of such equations and their solutions, and you should notice that there's a finite number of them (equal to the degree of the equation; e.g. a second-degree polynomial has two complex roots, third-degree polynomial has three etc.). And these roots are related to each other: if you take first of them and multiply by itself, you'll get the second root. Do it again, and you will get the third one. Repeating it will always give you the next root in a row. Until you make a full circle and end up where you started. Therefore they make a cyclic group. In the complex plane, they're usually located at the vertices of a regular polygon. You can do the same with any other of the roots of a polynomial. The bigger angle they make with the real axis, the more they "jump" in a rotation. E.g. if you multiply by the second root, you make twice the rotation of the first root in each step, jumping every other root. Multiplication by the third root jumps over three roots at a time. But no matter which one you choose, you always end up in one of the roots of your polynomial. No other number can be reached this way.

[@qaziarshad939]

How i can get all your lecture of Group theory, Real Analysis and Topology etc??????

[@whatsaapstatussong4782]

we also love you............and your videoes.

[@kunslipper]

Thank you so much.

[@soroushfth7055]

really great

[@Rsingh1]

Veey well explained

[@rxctr]

Thank you !

[@manjulacn3289]

Ma'am plz do videos on centralizer problems

[@shuhanchen1028]

My friends will never talk to me about mathematics

[@FrankChirwa-yq1oh]

Very nice

[@poulamikarmakar3759]

Mam ... Will you please upload some videos on nZ/Zn group... And a deep explained on it with definitions and examples... Please mam please

[@govindarajkhadka8160]

What will be the upto isomorphism of abelian group of dihedral D4 of order 8 ?

[@saurabhsingh-ow7ue]

thank you madam.

[@Sahilkhan-vj8tx]

Please give me some example of mixed group , torsion free group with examples.

[@ragavanm6416]

Tq mam. Very nice short lecture

[@MuhammadUsman-uu6rc]

where are you mam now miss you.. your lectures

[@ThePhinista]

This makes good sense.

[@nameisankur]

I like the shirt at end