Cyclic groups and generators

Question

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?

Answer 1

Its a group with two elements and therefore must be isomorphic to $\mathbb{Z}_2$ a simple check will show you which one is not the identity element and therefore the generator of the group.