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Order of an Element in a Group - Solved Examples

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#1

We provide these examples to help you get started.

Problem 1. Let be a group, such that . Find the order of .

Solution

From .

Proceeding in this manner (You should prove this with a lemma), we find that , positive integer.

Thus









But

So , and therefore, .


Problem 2. Let be a group, such that . Prove that .

Proof

We have seen that , for positive integer.

We are looking for the smallest integer, such that

Now



But

So , and therefore, .
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