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. What are Bijective Functions?: https://youtu.be/fcoC7gm4je8 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: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Abstract Algebra: https://amzn.to/3IjoZaO Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon! Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured