Cyclic Groups, Generators, and Cyclic Subgroups | Abstract Algebra

We introduce cyclic groups, generators of cyclic groups, and cyclic subgroups. We discuss an isomorphism from finite cyclic groups to the integers mod n, as well as an isomorphism from infinite cyclic groups to the integers. We establish a cyclic group of order n is isomorphic to Zn, and an infinite cyclic group is isomorphic to the integers. Finally, we introduce cyclic subgroups, and show the powers of an element will always form a cyclic subgroup. #abstractalgebra #grouptheory
Every Cyclic Group is Abelian: • Every Cyclic Group is ...
Every Subgroup of a Cyclic Group is Cyclic: • Every Subgroup of a Cy...
A Cyclic Group of Permutations: • A Cyclic Group of Perm...
Abstract Algebra Course: • Abstract Algebra
Abstract Algebra Exercises:
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[@ThefamousMrcroissant]

Another wonderful lecture. Second time I stumbled upon your content through my recommended feed. Pretty sure I already commented on your other video but you are a real standout among teachers in how coherent and neat your lectures tend to be. Solid script, good pacing, clear articulation. Keep up the excellent work!

[@WrathofMath]

Thanks so much! Let me know if you ever have any questions!

[@verrokade1993]

Perfect explained! Thank you soo much!!

[@2kreskimatmy]

i just love your videos, thank you for this

[@WrathofMath]

Thanks for watching!

[@erinmeyers-t9w]

Thank you for your videos, they really help me grasp these topics!! I wondered if you could please do a video on cyclic subgroups, the greatest common divisor, prime numbers, and all that mess? It's very confusing to me!

[@Laura-vj9zd]

Thank you for this video :) it helped a lot

[@WrathofMath]

Glad to hear it! Thanks for watching and check out my Abstract Algebra playlist if you haven't! czcams.com/play/PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN.html
Let me know if you have any video requests!

[@Eis461]

Thank you

[@WrathofMath]

Welcome!

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I've always said this, you are the best

[@WrathofMath]

Thank you, I work very hard on these 🙌 It’s summer now, I hope to make significant progress on producing more videos for my core playlists, and hopefully Wrath of Math will grow very healthfully this Fall!

[@MrCoreyTexas]

Challenge at 2:36 is 5, 5 generates 5,2,7,4,1,6,3,0 then repeats

[@springvibes5344]

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[@omerby.]

He said Amazon in another comment

[@moneyhoney_1]

Where did you get that trigonometry hoodie

[@WrathofMath]

Amazon!

[@standarddeviation7963]

Where are you getting 10, 5, and 8? 2:26

[@WrathofMath]

I am adding 3 (mod 8) repeatedly. So 7+3 = 10. But 10 mod 8 is 2. Then 2+3=5 and 5+3=8, but 8 mod 8 is 0.

[@eduyantra8048]

How will you generate negative integers using the generator ?? Please help

[@respectfullysammy]

I believe that should be a^-1. For example 1^-1 = -1, 1^-3=-1 (-1×-1×-1=-1). That should be an alternating group. I'm open to corrections tho

[@eduyantra8048]

@@respectfullysammy i referred a book in which, it was written, 1^-3 = (-1-1-1) =-3
Thank you for responding..

[@jeremymorgan1494]

1:22 - what is the definition for a generator under addition? this is so confusing to have the definition for G, and then immediately a new concept that's not defined is presented

[@jeremymorgan1494]

1:47 like saying Z= makes no sense given the definition that G={a^n : n in Z} (every element in G={1^n : n in Z } is 1). Where is the generator for addition defined

[@hschindele]

For a generator under addition it would look more like adding ‘a’ n times. So contains all possible sums of 1 and it’s additive inverse (-1) which clearly generates all integers.

[@PhdScholar249]

Sir please share some examples of Cyclic groups like Z10 Z15 like

[@WrathofMath]

Thanks for watching, and will do, just give me some time! Looking do make a permutation group lesson next.

[@paulfoss5385]

@@WrathofMath It might be neat to have a video explaining why if A generates a cyclic group G, then a function that multiplies every element in G with A is an automorphism of G. I bet you could work examples like Z10 and Z15 into such an explanation, and you would be laying the ground work for talking about primitive roots. Just a thought.

[@PhdScholar249]

Sir I'm your Subscriber

[@WrathofMath]

Thanks for watching!

[@PhdScholar249]

@@WrathofMath
Thanks a lot Sir

[@PhdScholar249]

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[@djehutisundaka7998]

(2^0) = SU(1)
(2^1) = U(1)
(2^2) = U(2)
(2^3) = SU(3)