let p be a prime. Show that
and show that
In general, .
It's not trivial at all that is cyclic. It's not very difficult to show once you know it, but you are probably not expected to come up with the proof. Instead you are probably expected to use this as a fact.
The second one is false - you have .
Notice that an homomorphism when is cyclic is completely determined once you know the image of a generator. Knowing this is determined by , but if is to be an automorphism then must be a generator ie. a nonzero element, but we have such elements. Now take automorphisms then if then so we can identify an automorphism with it's value at and this defines an isomorphism between