Prove that&nbsp;n∣ϕ(an−1)&nbsp;in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses&nbsp;Aut(G)

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Slade Higgins

2022-04-29

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

morpheus1ls1

Beginner2022-04-30Added 22 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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