Isomorphisms (Abstract Algebra)

An isomorphism is a homomorphism that is also a bijection. If there is an isomorphism between two groups G and H, then they are equivalent and we say they are "isomorphic." The groups may look different from each other, but their group properties will be the same.

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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

Milne, Algebra Course Notes (available free online)

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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison

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[woahitsben]

the quality of this video is incredible, the audio, the visuals, the pacing, the material, and the delivery

[Kaje_]

The best intuitive description of an "ismorphism" is to think in "analogies". Yep, an analogy itself is a good analogy for an isomorphism, you take some relationship and you change the context while maintaining that relationship in order to elucidate some property of the relationship. Of course, this is a very informal way to describe this. But it's a good intuitive insight.

[Socratica]

Our latest abstract algebra video is on *isomorphisms*!  These are functions which tell you when two groups are identical.  This is key, because the same group can appear in different places in wildly different guises.

(You can also have isomorphisms between rings, fields, modules, etc.  We'll cover those in separate videos.)

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[---gikf]

Wow! I understand isomorphism now. This is the best explanation. Thank you :)

[lynettemojica]

Thank you for this playlist... my friends and I are studying Abstract Algebra this summer before the class in the fall.

[rubempacelli]

I have no idea what Socratica is. I just stumble upon this wonderful video and I just want to say: thank you! This video is awesome! So well explained!

[alejrandom]

This is gold, I can't believe this series is free

[toasteduranium]

I’m too lazy to sit down and read a textbook sometimes. This engaging format also lends more memorability. I appreciate your demeanor! I’ve been looking for good abstract algebra resources for a while, and I think I’ve found what I needed.

[mountain]

A lot of things clicked into place for me after watching this video. Thank you for so concisely expressing these concepts!

[kemaltezerdilsiz]

I would like to really thank you for these videos. I am impressed by how well each concept is explained.

[MattRichards]

I'm really excited about this concept! Isomorphisms must be such a powerful tool to translate one type of group that can't be manipulated easily into a simpler one.

[danielberkowitz]

Best explanation for isomorphism I ever heard. Thank you so much!

[raymangoel]

The beauty of mathematics is in simplicity of seemingly complex ideas .... thank you a lot !!! for unveiling this treasure💝💝💫

[riturajsingh]

Wow, effective way to understanding. I appreciate you.

[Rishabh_Joshi_]

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[rapturian]

your channel and the presenter of these video series which is called "Abstract Algebra" are magnificent. I'm glad that I have you, guys. Also, I hope you'll continue your videos.

Take

[AnastasisKr]

You should have used the definition of isomorphism as a morphism with a left and right inverse. Then give the intuition that a homomorphism maps group structure to an object and the inverse maps back from it, the existence of the two sided inverse would then necessitate the structure can be moved freely back and forth between the objects.

This definition is not only equivalent in the case of groups, but it generalizes and unifies most mathematical objects. For example, you could draw the analogies with a familiar analogue: isomorphism of sets (ie: bijection), a visual/geometric analogue isomorphism of topologies (ie: homeomorphism) and then conclude by saying this concept (formed in this way) is the notion used in all of modern mathematics (ie: make a reference to category theory where the idea belongs).

Personal Comment:
- The set based definition you gave is a dated point of view which conceals elegant and intuitively simple mechanism by which the isomorphism preserves the structure of the group and is weighed down by set theoretic conceptual obstructions.

[AbhishekBhal]

Hi please do a video on cyclic groups... thanks

[sirelegant]

These videos are just superb, thank you Socratica

[jadekan]

Excellent! This helps me to understand isomorphism for the first time after school lecture! Thank you so much!