Question

If G is a finite cyclic group of order n, then determine autG.here G denotes the set of all automorphism of G.

Answer 1

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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)

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