I have the following question:
Let $ (\mathbb {Z} _4, +) $ be a cyclic group, such that $ \mathbb {Z}_{4}= \{0,1,2,3\}$. Find the generating elements of $ \mathbb {Z}_4 $.
The feedback says that the answer is 1 and 3, but I don't understand how I should proceed to find that result. I thought it was just using the powers ($1 ^ 0, 1 ^ 1, 1 ^ 2$, etc.), but I can't go on from there.
The group is small enough to test each element by taking successive powers; since the group is additive, this amounts to taking multiples of the elements.
In general, though, as a hint, look at the greatest common divisor of the order of the cyclic group and the representative of the equivalence class that is each element of the group.