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Today we will revise some basic Group Theory. Let be a group and . Assume that has finite order . Find the order of where is an integer.

Answer: , where .

Proof:

Our strategy is to prove that is the least smallest integer such that .

Now, we have . Note that is an integer and thus a valid power.

Suppose to the contrary there exists such that .

Since has finite order , we have , which leads to . Note that and are relatively prime.

Thus , which implies that which is a contradiction. This proves our result. ðŸ™‚