Proof of Cayley's Theorem | Abstract Algebra

We go over how to prove Cayley's Theorem, which states the incredible fact that every group is isomorphic to some permutation group. This proof will require us to construct a group of permutations of G, then show that G is isomorphic to this permutation. There are a fair bit of details to work out, but it is a surprisingly straightforward proof. #abstractalgebra Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join Permutation Groups: https://youtu.be/NwqCi63p2ik Isomorphic Groups: https://youtu.be/fGqx_-F7zN4 Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz Get the textbook for this course! https://amzn.to/3IjoZaO ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Outro music by Ben Watts and is available for channel members. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath 0:00 Intro 0:48 Constructing Permutations 3:17 Making the Permutation Group 6:23 The Isomorphism 9:02 Conclusion