Group Isomorphisms: Definition and Examples

In Abstract Algebra, given two groups G and G̅, an isomorphism from G to G̅ is a function φ: G → G̅ that is one-to-one, onto, and operation preserving (φ(ab)=φ(a)φ(b) for all a,b in G). Example isomorphisms include: 1) an exponential function φ: (ℝ,+) → (ℝ+,*) defined by φ(x)=2^x, 2) for an infinite cyclic group G with generator "a", φ: (G,*) → (ℤ,+) defined by φ(a^n)=n (canonical mapping), and 2) for a finite cyclic group G with generator "a" of order n, φ: (G,*) → (ℤn,+) defined by φ(a^m)=m mod n (canonical mapping). Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter: https://twitter.com/billkinneymath 🔴 Follow me on Instagram: https://www.instagram.com/billkinneymath/ 🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.