Definition of the Order of an Element in a Group and Multiple Examples 

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Notice f(a)^n = f(a^n) = f(e) = e' where e' is the identity of the Group f is mapping onto. Now, suppose ord(f(a)) = m < n. Then, f(e) = e' = f(a)^m = f(a^m). Now, since f is injective, e = a^m which is ofc a contradiction.