# Prove that n|\phi(a^n-1) in "Topics in Algebra 2nd Edition" by

Prove that $n\mid \varphi \left({a}^{n}-1\right)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $Aut\left(G\right)$

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