abstract algebra one question

abstract algebra one question

I want a detailed step by step proof for the following abstact algebra question (just one question.) Please do not use any theorem that do not belong to the chapter

here and please read the question carefully to make sure that you know how to answer it. Let me know if you have any questions. Thanks!

1. If f ∈ A(S) has o(f) = p, where p is a prime number, then show that for all s ∈ S O(s) = {f k (s)| k ∈ Z} contains either one element, or p elements.

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