Question

If g is a finite cyclic group of order n, then determine autg. (here g denotes the set of all automorphisms of g.)

Answer 1

Thank you for asking this question. Here is your answer:

Let G = <a>, where o(a) = ∣∣G∣∣ = n

Let φ∈ Aut (G)

o(ak)=o(φ(a))=o(a) . … (⋆)

o(ak)  = n/gcd(k,n). So k∈{m:1≤m≤n,gcd(m,n)=1}

Aut (G) = {φ:φ(a)=ak,gcd(k,n)=1}.

Aut (Zn) ≅ U(Zn)

|Aut(Zn)|  = ϕ(n)

If there is any confusion please leave a comment below.