Isomorphic Groups and Isomorphisms in Group Theory | Abstract Algebra

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.

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

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[mcmoodoo]

The author clearly has a skill of providing clear explanations! Well done, sir!

[KermitTheHermit.]

Understood perfectly! Thank you for a different perspective. Was stuck with the textbook definition for long. Thanks again🙏🏻😊

[wenzhang]

Please keep up the good work, thank you!

[VijitChandna]

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 :)

[lamyamalcolm-uuzp]

Very skillful and talented, thank you so much. You videos help me a lot with my studies here.

[claytonbenignus]

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 . . .

[gabriels.i.]

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!

[MrCoreyTexas]

For anyone interested, you should look up the Wikipedia on the Klein Group with 4 elements

[JTan-fqvy]

Thanks for the great video! Is there any theory that deals with the generalization of this isomorphism? For example, if I want to verify an equivalent relation between two mathematical objects with arbitrary properties (not specifically the ones of binary operator for groups), is there a modification of the definition in time 1:50 that can give a generalized notion of isomorphism?

[alexdrougkas]

Thank you so much for the lecture. Keep up the great quality of work!

[jaaaaaaaaaaaac]

very nice video！you should put this into your list, can't find this one in the list.

[kabirbhattacharyya]

One thing I was wondering.... How does isomorphism "transforms" a group to another? Like how I thought.... It's like a bridge to
Each storeys of two almost identical buildings. One is red, one is blue.. etc. what exactly I'm calling the "Isomorphism"?
Also could you help me with the "transforms" a group to another?

[ahasdasetodu]

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?

[madomalene]

For the first example, how did you obtain the second table. What rules were you using to perform the multiplication

[gp]

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?

[kreskimatmy]

do you get into Cayley's theorem in some video?

[VijitChandna]

The chapters seem to say homomorphism for some reason

[algierithm]

Hello what notepad are you using? Thanks

[scito]

so isomorphism is a homomorphism that is a bijection
right?