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1 $begingroup$ 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 share | cite | improve this question

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I'm learning how to use Spring JPA and am have configured my database to successfully add elements to my database. I'm just having trouble formulating a query to retrieve my results (a list of Employee, filtered by lastname): //Employee entity: @Entity public class Employee { @Id @GeneratedValue(strategy=GenerationType.IDENTITY) long id; String firstName; String lastName; public Employee() { } public Employee(long id, String firstName, String lastName) { super(); this.id = id; this.firstName = firstName; this.lastName = lastName; } p