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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, 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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