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

Slade Higgins
2022-04-29
Answered

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

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morpheus1ls1

Answered 2022-04-30
Author has **22** answers

Step 1

Note that$\varphi ({a}^{n}-1)$ measures the number of automorphisms of $\frac{\mathbb{Z}}{(an-1)}\mathbb{Z}$ .

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.

Note that

There is a subgroup of order n in this group: if

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