Group Theory Proof: The order of x is the order of x inverse  Summary and Q&A
TL;DR
Proving that in a group, the order of x is equal to the order of x inverse, whether x has finite or infinite order.
Key Insights
 In a group, the order of an element x is equal to the order of its inverse.
 The proof is done in two cases: finite order of x and infinite order of x.
Transcript
let g be a group and x an element of g you're going to prove that the order of x is equal to the order of x inverse proof let's do it in cases so case one this will be the case where x has finite order so x has finite order so since x has finite order let's go ahead and let n be equal to the order of x well then this means that x to the n is equal ... Read More
Questions & Answers
Q: How is the order of x related to the order of x inverse in a group?
In a group, if x has a finite order n, then the order of x inverse will also be n, showing that the order of x is equal to the order of x inverse.
Q: What happens if x has an infinite order in a group?
If x has an infinite order, the claim is that x inverse also has an infinite order, establishing that the order of x and x inverse are equal in the case of infinite order.
Summary & Key Takeaways

Proving the order of an element x in a group is equal to the order of its inverse in two cases: finite and infinite orders.

Case 1: For x with finite order n, showing that the order of x is equal to the order of x inverse.

Case 2: For x with infinite order, demonstrating that the order of x and x inverse are both infinite.