We have whenever

Thus:

and

So it's enough to check that but this is true since both are cyclic groups of the same order.

is cyclic if and only if where p is an odd prime.

In your particular case you can check that and ...

The reason why you were told that what you did was not enough is because you have to consider the "multiplication" between distinct elements... - if you found a function and showed that it is a bijection and you'd be done...