Prove if |a| = n in a group, then ⟨a^i⟩ = ⟨a^j⟩ iff |a^i| = |a^j| iff gcd(n,i) = gcd(n,j)

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Prove if |a| = n in a group, then ⟨a^k⟩ = ⟨a^gcd(n,k)⟩ and |a^k| = n / gcd(n/k) (ILIEKMATHPHYSICS) Prove if |a| = n in a group, then ⟨a^k⟩ = ⟨a^gcd(n,k)⟩ and |a^k| = n / gcd(n/k) (ILIEKMATHPHYSICS)
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