Proof: Finite Order Elements Have n Distinct Powers | Abstract Algebra

We prove that a group element x of order n has n distinct powers, namely x^0, x^1, ... , x^(n-1). To do this we first prove that all powers of x are contained in the aforementioned list, and then we prove all powers of x in that list are distinct. #abstractalgebra #grouptheory

Order of Group Elements:    • Order of Elements in a Group | Abstra...  
Finding the Order of Group Elements:    • Finding the Order of Group Elements |...  
Infinite Order Elements have Distinct Powers:    • Infinite Order Elements have Distinct...  

Abstract Algebra Course:    • Abstract Algebra  
Abstract Algebra Exercises:    • Abstract Algebra Exercises  

◉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:   / 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:   / wrathofmathedu  
● Facebook:   / wrathofmath  
● Twitter:   / wrathofmathedu  

My Math Rap channel:    / @mathbars2020