To which group is the $\mathbb Z_{20}^*$ isomorphic? I have a question saying that to which group is $\mathbb{Z}_{20}^{*}$ is isomorphic, where $\mathbb{Z}_{20}^{*}$ is the set of the not zero divisors of $\mathbb{Z}_{20}$.
Here is what i think: $\mathbb{Z}_{20}^{*}=\{1,3,7,9,11,13,17,19\}$. It has 8 elements. Then it is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ or $\mathbb{Z}_{8}$. But how can I decide which of them? And is this group a multiplicative group or an additive group? How can I know?
Thank you

To hint your last question first, if $a,b$ are each relatively prime to $20$, then $ab$ is also relatively prime to $20$.  $a+b$ need not be relatively prime to $20$.
To hint your first question, $\mathbb{Z}_8$ has an element of order 8,  $\mathbb{Z}_2\times\mathbb{Z}_4$ has an element of order 4 but no element of order $8$, while in $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ all elements but the identity have order 2.
