Cyclic groups and generators For each of the groups $\mathbb Z_4$,$\mathbb Z_4^*$ indicate which are cyclic. For those that are cyclic list all the generators.
Solution
$\mathbb Z_4=${0,1,2,3}
$\mathbb Z_4$ is cyclic and all the generators of $\mathbb Z_4=${1,3}
Now if we consider $\mathbb Z_4^*$
$\mathbb Z_4^*$={1,3}
How do i know that $\mathbb Z_4^*$ is cyclic?
In our lecture notes it says that the $\mathbb Z_4^*$ is cyclic and the generators of $\mathbb Z_4^*$=3
Can anyone help me on the steps to follow in order to prove the above?

Consulting here, you can easy see that in modulo $4$ there are two relatively prime congruence classes, $1$ and $3$, so $(\mathbb{Z}/4\mathbb{Z})^\times \cong \mathrm{C}_2$, the cyclic group with two elements.
