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We prove that , also known as (easier to type).

Define by .

First we show that it is a homomorphism:

Next we show that it is injective:

Thus, .

Let .

.

Thus, the only automorphism that maps to 1 is the identity.

Thus, is trivial.

Finally, we show that it is surjective.

Let . Consider such that , , , …, .

We claim that is an automorphism of .

Firstly, we need to show that . This is because . Hence if is the order of , i.e. , then , which implies that which implies that is at least . Since the order of is also at most , .

Finally, we have and thus we may take as the preimage of .

Hence is surjective.

This is a detailed explanation of the proof, it can be made more concise to fit in a few paragraphs!

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