Cyclic Groups (Abstract Algebra)

[@Socratica]

If you'd like to learn more, we have a free course on Group Theory! www.socratica.com/courses/group-theory

[@IamRigour]

Doesn't the set of all nth roots of Unity, a subset of complex numbers with modulus of 1, form a group whose operation is multiplication? Also, isn't this group cyclic? In Abstract Algebra (Theory and Applications) by Thomas W. Judson, it is mentioned that the set of all nth roots of Unity forms a cyclic group.

[@angelxmod3]

@@IamRigour It is just isomorphic to these integer cyclic group so that is why she said they are the only ones

[@IamRigour]

@@angelxmod3 Yeah, I've studied it more. I understand now.

[@erodotosdemetriou6506]

So thankful! In 5 min, you explained a one-hour no-sense university presentation! Not everyone with a PhD should become a professor. Some people are charismatic in explaining things while others are not so much!

[@karlalopez6503]

Phd don't take classes to become teachers 😅 but k-12 teachers have to take classes to learn how to be good teachers and they are evaluated in it ... but phd is hard to get 😢 but it would be nice to have good university teachers all the time , I don't even understand the notes 😂😂

[@vanguard7674]

I wish a million subscribers on this channel

[@Socratica]

You are so kind, thank you! Our dearest wish as well! :) We'll keep making videos and hopefully the viewers will come. Everyone share with your friends!

[@joehudson440]

Socratica I admire your level of sophistication in your presentation of Abstract Algebra. I feel that I have a sense of what a group means. Thanks.

[@MegaKaran96]

Me too✌️✌️

[@ValeriaCalarco]

GREAT! THANKS SO MUCH👍👍

[@jacobguerreso675]

Almost there!

[@e.jhumphrey9893]

Thank you Socratica! Your videos are wonderful and have given me new determination to continue my abstract algebra course. One of your greatest strengths (for me personally), is having well explained examples to accompany all your definitions and explanations. Most lecturers at the Tertiary level just tend to spout off theorems and definitions without actually showing any applicability to what we are learning- so this has completely changed the way I understand my maths. I will definitely be sharing this with all of my peers. Thank you again :)

[@KenPryde]

I can't adequately express how fantastic this video series is, both in quality of the material, and the clear, informative, and accessible presentation.

[@DavidKotschessa]

This is very well done. I would not be terribly upset if you decide to do a series on algebraic topology.

[@Trunks47r786]

Don't you think that's a big jump?

[@DavidKotschessa]

Trung Nguyen Maybe, but there's a lot of material on point set already out there. I often feel that algebraic topology is explained in an obscure way and that this team could really tackle it.

[@ninosawbrzostowiecki1892]

I would be very upset.

[@traciyandell8297]

That would be wonderful! I've avoided topology.

[@mr.p2665]

Motaimotararo

[@ideedrafiqi410]

Thanks Socratica!, I aced my Group Theory majors.

[@Socratica]

That's wonderful news!! Congratulations!!

[@SuperOpposum]

This is such an amazing playlist, making it all so simple and clear while doing it so much faster than my professor. I immediately checked what other subjects you have, and it seems like I'll be using you guys next semester as well, so pleased.

[@Socratica]

We're so glad you've found us!! Keep us posted about your progress! 💜🦉

[@traciyandell8297]

A super channel. I'm using the Abstract Algebra section to get me through a class. The instructor is great! She adds humor and understanding of the concepts -- presents very well. Thanks for this channel!

[@Socratica]

We're so glad you've found us! Thanks for letting us know the videos are helpful for you. That really inspires us to keep making videos! 💜🦉

[@kellybrower301]

The diagram for Integers mod n under addition has shown me the light. Thank you gorgeous stranger with the voice of power and mind of algebra.

[@Babubro]

Fairly speaking I accidentally watched your one video, Now here iam watching each and every video with out a single skeep❤❤❤❤❤ Great mam, Lots of love from Nepal

[@havock0701]

I love this series! It helped me a lot preparing for my Abstract Algebra exam!! Please try to make series for Real Analysis and Differential Geometry if you can! Not many well produced stuff for upper division math courses on KZbin.

[@paulfrazier1021]

I have yet to take an actual class on abstract algebra (been studying it on my own because I find it fun), but these videos really fill in the holes of my understanding. I knew a fascination with axiomatic set theory would be useful someday!

[@salounik.2894]

Everything was good. I have my exams tomorrow and I found this good piece. I'm so happy. And oh, that music at the end...lovely!

[@deckfart1560]

Your videos have been really helpful , especially nowadays when we can't go to our Universities anymore. Thanks a lot.

[@Socratica]

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

[@tearman005]

NO

[@alongimchen8127]

I accidentally clicked in this video and the way you teach is absolutely amazing...I've already subscribed your channel..I would kindly request you to upload more examples and videos on group theory .. thank you Ma'am

[@omkark7597]

Explanation and content is really perfect. One suggestion for abstract algebra videos if you can explain each concept of group with diagram... like small representative model then it will help us in complex concepts. thanks a lot for videos. Appreciated all your work.

[@moularaoul643]

It hurts me to see these wonderful videos not reaching millions of views !!! Please can you also do some lessons on topology, complex analysis and differential geometry? Please!!!!

[@howmathematicianscreatemat9226]

This is incredibly powerful teaching! Like this 95% of Americans could study and graduate in university Mathematics, my deep respect!

[@Aycore2011]

Ncaaawww !! how cute, an entire group in "H" that is all about "X" ...so adorable and its small *sheds tear*.

[@huutaiphung9559]

It's just so great, hope you will make more videos in Abstract Algebra

[@aswathik4709]

your presentation is wonderful and to the point!!

[@bazislapper6283]

Im so glad, that i noticed this channel, its amazing content

[@NikesDarkslayer]

amazing channel, makes me eager for knowledge

[@Socratica]

Your comment made us smile so much! Thank you for sharing our love of learning new things!

[@Jat-Jat]

These videos are helping me pass Abstract Algebra I swear.

[@Socratica]

This is so wonderful to hear, thank you for letting us know our videos are helping! It really inspires us to make more videos. Good luck with your course and let us know how you get on! 💜🦉

[@waseemullah7022]

Bundle of thanks this is very helpful lecture

[@julianarodrigues1861]

Thank you!! This topic is more clear for me now

[@Socratica]

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[@mortervolk6676]

Love your work! Keep it up, guys. You're the best! (Math enthusiast from SYRIA)...

[@馬陸]

That is a great idea to have a beautiful face and voice person to teach a boring subject! That makes sure viewer not easy to fall asleep and keeping one awake.

[@SphereofTime]

1:15 What's the smallest cyclic group? 2:14 why y is the smallest?

[@SphereofTime]

4:00 finite, infinite integer cyclic group

[@sanskarsahu2177]

well even though I don't have this under my academic portion. I love watch your videos on this topic ,and any other topic too ,great job ,keep it up ☺

[@khadijehalrababah3996]

واو الشرح جدا رائع ومميز ما كنت فاهمه على دكتوري لما شرح الموضوع بس لما حضرت الفيديو اكتشفت انو المفهوم جدا سهل وبسيط ❤❤

[@thomasfisherson]

Thank you for providing an easy way to understand this stuff. Honestly this stuff isn't hard - you could probably teach it in high school actually. But everything online reads like you already need a BS in math or something.

[@athelstanrex]

I think that the point of this series is to introduce modern algebra. If you just watched this series to pass abstract algebra, you would most definitely fail.

[@zackm5693]

Great series, doing one on topology would be great too

[@aaryabhatia2478]

Wish I found this series sooner!

[@levk2027]

thanks socrtica,,thanks to you I don't hate abstarct algebra any more,,,wish you all the best,,,keep on doing such incredible and understanding videos,,,

[@himangshukalita3099]

I like the way you explain things without any bullshit..you are nice and direct. Thank you.

[@franciscomelojunior2535]

Thank you so much for this video, merci beaucoup.

[@giulianobarrellagarcia3205]

you're seriously awesome, thank you for the video and I hope that you keep posting!

[@SunnySingh-cd1rr]

It’s a perfect explanation that I wants... well done..Keep it up.we need more from you.. May you get 1million subscribers soon..

[@Mahesha999]

What do you mean by "complete collection" of cyclic groups at 3:44 ? There can be more cyclic groups defined on other operations right?

[@ang9585]

it's always crystal clear

[@luyombojonathan6688]

Beautiful explanation

[@godsowncoders8272]

Me : Abstract Algebra is difficult Socratica : Hold my Beer!

[@mahmoudelsayed6269]

i wish you complete this series this series is very good thank you

[@amath7874]

Thank you very much mam, I understood it very well.

[@Himalayan_Guy]

I am loving video series on algebra, please make some more videos on other domanis of pure mathematics.

[@AnchalShukla-q4w]

Hill ☺️

[@AnchalShukla-q4w]

I am anchal

[@AnchalShukla-q4w]

And you

[@AnchalShukla-q4w]

i belong in India

[@senahdongasso4580]

I don't understand well English but this video help me very much. Good job

[@sharavanakumar2737]

Amazing channel. Very very Thanks Socratica it is because of you I am able to obtain good marks in Group theory....

[@ramanujaninstitutecsirnetj6999]

nicely explained. concepts are very clear.

[@gagan9124]

really helpful and easy to understand

[@32-rishavsharma39]

What an explanation. Thank you🙏

[@EmmanuelGidudu-k3t]

great and nice explanations, i can comfortably explain a concept in the absence of my lecturer

[@peterxie7739]

real nice video series.Really Hope to see some topology videos.

[@vrendus522]

Your'e a sister figure to me. Older sister was like you.I need her again, so am intently studying what ur saying. Thanks, best to you and yours.

[@r3dw0lf31]

One of the best channel...

[@sofiasarigiannidi8708]

This is very well done but I wish there was more talk upon the definition of a generator and also that the cyclic groups are the only ones up to isomorphisms

[@supposexy]

Outstanding Method!!!!! Love it

[@anushkadixit2593]

U r great. I finally understand cyclic group😘 Thanku

[@MattMooreTrails]

Very, very, very helpful. Thank you.

[@himangshukalita3099]

your contents really help.

[@argonwheatbelly637]

Math is an Art, and abstract algebra is poetry. Treat Math as only a Science, and you will relegate it to the utilitarian. Treat it as an Art, and you will begin to converse in a language that transcends common communication.

[@wancho919]

Las explicaciones son bastante claras, me gustaría que subiera un vídeo en castellano. Gracias por el gran aporte en el aprendizaje del álgebra abstracta.

[@ChaudharyAteeq440]

Please upload more videos on Pure Mathematics, Thanks...

[@inammwt4571]

Dear Ma'am I like the way you explain, I also like your look and voice

[@mypaldan]

@2:06 why does she say "in order to be a group it must contain all positive and negative values of y"? Isn't y, 0, -y a group under addition?

[@MuffinsAPlenty]

No, it isn't closed under addition, since y+y must be in there if y is in there.

[@VinodThakur-i4i]

Thanks 😊

[@2bsirius]

For more detail on this check out a video playlist from Harvard: Abstract Algebra/Group Theory. It has a great set of lectures by Harvard math prof, Benedict Gross which incorporate cyclic groups as well as many other concepts in abstract algebra.

[@abdelhamid_moukrime]

Thier is anothter importants cyclic groups like: {-i,-1,i,1}= {-j,j1}= j=exp(2iπ/3)

[@helloitsme7553]

3:45 what about the powers and inverse powers of any number c under multiplication? It can be created using only c and it's repeated under multiplication, the identity is in there, the inverse is in there...

[@tcpjh]

I really love your channel!

[@MegaKaran96]

Hey thanks!! That's was great help for me to learn basics of abstract algebra.I would say this is the series everyone watch after 3blue1brown's essence of linear algebra.

[@modolief]

Nice vidoes! Brings back memories ...

[@sahanigovind6939]

very nice explanation

[@MiddleClassTalk]

not need to see any more video on groups after this on...nice video!

[@ggopal453]

Thanks socretica from India

[@MuhammadUsman-uu6rc]

Its a beattful video mem.. I show your videos .. And present it into school

[@rmschindler144]

is ‘SICK-lick’ really the way mathematicians pronounce it? . unfortunately I don’t recall it from my years at Waterloo studying math, but I’m reasonably sure that there we said it like ‘SIGH-click’ . (cf. the pronunciation of ‘cycle’)

[@BlahBlahBlah13623]

These are amazing. Please make more. I would even pay (or turn my add block off if it were a requirement!)

[@EpiCuber7]

How do you know those are the only cyclic groups? What about, for example, the even integers = ? Or any examples with multiplication (such as the group of all integral powers of 2)?

[@MuffinsAPlenty]

They are the only cyclic groups *up to isomorphism.* If you find another cyclic group, it is definitely isomorphic to Z or Z/nZ for some positive integer n. So for example, the integers are isomorphic to the even integers as groups. The isomorphism is given by f : Z → 2Z where f(x) = 2x. Similarly, the integers are isomorphic to the integral powers of 2. Use the function f(x) = 2^x.

[@MartinGenter-dp5nx]

Danke!

[@Socratica]

Goodness, thank you so much, kind Socratica Friend!! 💜🦉

[@MartinGenter-dp5nx]

@ amazing lectures, a great pleasure to watch

[@slashpl8800]

Amazing explaination, thank you!

[@rasraster]

Isn't it more correct to say that all cyclic groups are isomorphic to Z and Z/nZ, rather than saying that those are the only cyclic groups?

[@vanguard7674]

Btw, what about U(n) the multiplicative group of units mod n? That's cyclic too right?

[@Socratica]

Not always. For example, consider the units mod 8 under multiplication. They are 1, 3, 5, 7. Next, look at the group generated by each element: = {1, 3}. = {1, 5}. And = {1, 7}. So in this case, the units are isomorphic to the group Z/2Z x Z/2Z. However, there are many cases when the units *are* cyclic. For example, the units mod 5 are {1, 2, 3, 4} and this group, under multiplication, is generated by 2.

[@jean-patrickpelletier4162]

It's cyclic of order n-1 when n is a prime

[@joshh6922]

it's cyclic when your integer space and some k are relatively prime.

[@jaredronning3020]

There are many cyclic groups different from Z or Zn. The claim is that they're all isomorphic to Z or Zn.

[@thcoura]

Fascinating Subject

[@kautukraj]

Very helpful!

[@pascalfrei8178]

@2:11 so if H= ‹y› for some y, then the group H is equal to the subgroup, or no?

[@nitkonigde1381]

When you say that integer rings are the only cyclic groups, do you mean just in R? Could be cyclic groups in the complex plane that aren't integer rings? I think so but I don't know. Or are you saying that all cyclic groups are isomorphic to the integers?

[@aritot2011]

Shes saying that all of them are isomorphic to the íntegers or the integers modulo n for some n. Take, for example, the multiplicative group {1,-1,i,-i}. It is generated by i or -i, therefore its cyclic, and it is isomorphic to Z/4Z. You can check It by hand by the isomorphism that maps i^n to n.

[@zarmmraz5267]

um.. thanks for your video. Becoz your video help me to understand about 1.5 hours lecture video (cyber security and crypto)

[@navedm1424]

Could you please create a playlist on graph theory?

[@MG-im3vn]

keep on the awesome work. thank you

[@TOP3VERSE]

Great explanation mam..

[@voiceofnation7493]

Thanks, Ur the best in what ur doing...

[@2kreskimatmy]

0:02 exactly by one element, or by set of generators?

[@ranael-achkar9936]

You don't know how amazing you are!

[@KishoreReddyMath]

Well explained

[@matthewwhite8882]

Dig the sound effects...

[@Coolblueocean2001]

Great channel!