abstract algebra?

Question

list all isomorphisms from Z-8 toZ-8

Answer 1

Since ℤ₈ is cyclic, any isomorphism is completely determined by where it sends any generator of ℤ₈. So we can list them by where they send the element [1]. Also note that any isomorphism of ℤ₈ must send [1] to a generator of ℤ₈. There are exactly four such generators of ℤ₈ -- 1, 3, 5, and 7. So the isomorphisms are n ↦ n, n ↦ 3n, n ↦ 5n, and n ↦ 7n.