For a given group to be cyclic, does the generator have to produce exactly all of its elements?

Question

Take for instance $\Bbb Z_5^*$ under addition.

Take the generator of $\langle1\rangle=\{1,2,3,4,0,\ldots\}$, yes it generates the entire group but it has an additional $0$. Is this still a generator?

Answer 1

You are mixing two things here: the group $(\Bbb Z_5,+)$ and the group $(\Bbb Z_5\setminus\{0\},\times)$. Yes, the first group is generated by $1$. However, $1$ is the identity element of the second one, and therefore it does not generate it (the subgroup generated by $1$ is $\{1\}$). It generated by $2$ and $3$, but not by $1$ (nor $4$).