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

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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.