Prove that&nbsp;n∣ϕ(an−1). Any natural solution that uses&nbsp;Aut(G)

Step 1Note that&nbsp;ϕ(an−1)&nbsp;measures the number of automorphisms of&nbsp;ℤ(an-1)ℤ.There is a subgroup of order&nbsp;n&nbsp;in this group: if&nbsp;ϕ&nbsp;is the automorphism sending 1 to a, then&nbsp;ϕ&nbsp;generates a subgroup of order&nbsp;n. The statement follows from Lagrange's Theorem.

Question: "Prove thatn∣ϕ(an−1). Any natural solution that usesAut(G)"

College
Algebra
Abstract algebra

Hugh Soto

Answered question

2022-03-28

Prove that $n ∣ ϕ (a^{n} - 1)$ . Any natural solution that uses $Aut (G)$

Answer & Explanation

Esteban Sloan

Beginner2022-03-29Added 21 answers

Step 1
Note that $ϕ (a^{n} - 1)$ measures the number of automorphisms of $\frac{ℤ}{(a n - 1)} ℤ$ .
There is a subgroup of order $n$ in this group: if $ϕ$ is the automorphism sending 1 to a, then $ϕ$ generates a subgroup of order $n$ . The statement follows from Lagrange's Theorem.

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