What's the Secret Sauce of the Orbit Stabilizer Theorem Proof?

The Orbit-Stabilizer Theorem statement is: if G is a finite group of permutations on a set S, then |G| = |orb(i)|*|stab(i)| for any i ∈ S. How is this theorem proved? The conclusion is equivalent to the equation |G:stab(i)|=|G|/|stab(i)|=|orb(i)|, where |G:stab(i)| is the index of the subgroup stab(i) in G, by Lagrange's Theorem. This is the number of left cosets of stab(i) in G, which is the same as the number of elements in the factor group G/stab(i). To prove the Orbit-Stabilizer Theorem, we must mathematically construct a bijection T:G/stab(i)→orb(i) (so T must be one-to-one and onto). Given a permutation α ∈ G, the natural mapping is T(α*stab(i))=α(i) (map the left coset of the stabilizer in G containing α to the function output α(i)). The first step of the proof is to show that T is a well-defined mapping. In other words, if we have two different coset representatives, are their function outputs at i equal? That is, if α*stab(i)=β*stab(i), does α(i)=β(i)? 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.