Isomorphic Groups and Isomorphisms in Group Theory | Abstract Algebra
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- čas přidán 9. 07. 2024
- We introduce isomorphic groups and isomorphisms. We'll cover the definition of isomorphic groups, the definition of isomorphism, an example of isomorphic groups with a group table, we'll prove two groups are isomorphic, discuss how to show two groups are not isomorphic, and finish with a few theorems. In short, an isomorphism f from a group G to a group H, is a bijection from G to H such that for all a, b in G, f(ab) = f(a)f(b). We say the isomorphism "preserves the group operation". If an isomorphism exists between G and H, then G and H are said to be isomorphic. #abstractalgebra #grouptheory
In our first example we see how the group Z3 is isomorphic to a multiplicative group of 3 elements. In our second example we see how the group of real numbers under addition is isomorphic to the multiplicative group of positive reals. To do this we use the bijection f(x) = e^x.
What are Bijective Functions?: • Bijective Functions an...
Basic Properties of Isomorphisms: (coming soon)
Group Isomorphism is an Equivalence Relation: (coming soon)
Proof of Cayley's Theorem: (coming soon)
0:00 - What is an Isomorphism?
1:45 - Definition of an Isomorphism and Isomorphic Groups
3:45 - Further Explanation of Preserving the Group Operation
4:26 - Isomorphisms are Renamings
5:08 - Example with Group Tables
7:19 - Proving two Groups are Isomorphic
11:20 - How to Show two Groups are NOT Isomorphic
12:29 - Some Theorems
Abstract Algebra Course: • Abstract Algebra
Abstract Algebra Exercises: • Abstract Algebra Exerc...
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The author clearly has a skill of providing clear explanations! Well done, sir!
Many thanks!
Understood perfectly! Thank you for a different perspective. Was stuck with the textbook definition for long. Thanks again🙏🏻😊
Glad to help, thanks for watching!
First! Tommorow is my exam and I had commented on his channel about this topic and he sent me an unlisted link! Thank you so much :)
I Get It!!! You could say that if you take a Zane Grey Novel and transform a few words (Rancher's Daughter = Martian Princess; Rifle = Disintegrator; Stage Coach = Rocket Shoip; The Cavalry = Star Fleet; etc.), you get a Star Trek Episode . . .
Thank you so much for the lecture. Keep up the great quality of work!
Thank you Alex!
Thank you so much! Very clear and rich explanation. I would like to ask...Isomorphism seems pretty restrictive as a way to study identity/similarity between groups. Is there any concept in abstract algebra that can account for "weaker" forms of similarity? Thanks!
Great question and the answer is a big yes! czcams.com/video/rJpu22jMeIY/video.html&pp=ygUSaG9tb21vcnBoaWMgZ3JvdXBz
Very skillful and talented, thank you so much. You videos help me a lot with my studies here.
Glad to hear it, thanks for watching!
1:42
do you get into Cayley's theorem in some video?
The second theorem mentions a set of all groups but from my understanding of set theory such a thing would lead to contradictions the same way a set of all sets does. Wouldn't it be better to say a class of all groups?
For the portion where you discuss ways to find groups that are NOT isomorphic, you give 4 criteria but I'm curious what the difference between #2 and #3 are? If a G1 has an element of order n, does that not make it cyclic, which would be the same as #2?
Thanks for watching and for the question! Perhaps you're confused because you think I mean 'n' to be the order of the group? I simply mean n to be a finite number, and a group having an element of finite order does not force it to be cyclic. Does that answer your question?
@@WrathofMath Gotcha! It does answer my question. Thanks.
For the first example,how did you obtain the second table. What rules were you using to perform the multiplication
he just renamed all elements and the operation of g1. that is how he got g2. then he proved that these groups are isomorphic (the same), which is trivial since one is a renaming of the other
very nice video！you should put this into your list, can't find this one in the list.
In the playlist? Weird, I see it. It is right after Permutation groups and before Order of Elements in a Group! I have spreadsheets on spreadsheets to keep all my playlists organized haha!
Hello what notepad are you using? Thanks
Notability!
@@WrathofMath thanks much. btw, I love your videos.
The chapters seem to say homomorphism for some reason
Looks like they're correct in the description, will probably just take some time to update hopefully!
so isomorphism is a homomorphism that is a bijection
right?
if anyone wants to know
i asked Bing AI and it basically said yes
Exactly!