How many group homomorphisms from $Bbb Z_3$ to $text{Aut}(Bbb Z_7)$
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How many group homomorphism from $Bbb Z_3$ to $text{Aut}(Bbb Z_7)$ ? My attempt: Since $Bbb Z_3$ is cyclic, hence the homomorphisms are all determined by $phi(overline{1})$ . On the other hand $text{Aut}(Bbb Z_7)cong Bbb Z_6$ . Since $o(phi(overline{1}))mid |Bbb Z_3|$ , so $o(phi(overline{1}))$ can only be $1,~3$ . If the order is $1$ , it means $phi(overline{1})=overline{1}$ . It is indeed a homo. If the order is $3$ , it means $phi(overline{1})=overline{2}$ or $overline{4}$ . So the total number of group homomorphisms from $Bbb Z_3$ to $text{Aut}(Bbb Z_7)$ is three, as I show above. Am I correct?
abstract-algebra
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