Generators and cyclic group concept These statements are false according to my book. I am not sure why though


*

*In every cyclic group, every element is a generator

*A cyclic group has a unique generator.


Both statements seem to be opposites.
I tried to give a counterexample


*

*I think it's because $\mathbb{Z}_4$ for example has generators $\langle 1\rangle$ and $\langle3\rangle$, but 2 or 0 isn't a generator.

*As shown in (1), we have two different generators, $\langle1\rangle$ and $\langle3\rangle$

Take $Z_n$. This group is cyclic and the generators are $\phi(n)$ = all the numbers that are relatively prime to $n$
