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

Slade Higgins 2022-04-29 Answered

Prove that nϕ(an1) in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses Aut(G)

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Answers (1)

morpheus1ls1
Answered 2022-04-30 Author has 22 answers
Step 1
Note that ϕ(an1) measures the number of automorphisms of Z(an1)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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