What is this notation? Cyclic group $\mathbb{Z}^*_8$ $\mathbb{Z}^*_8$
As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8.
So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$?
I thought the $*$ meant excluding zero, so I was confused to learn that the elements of $\mathbb{Z}^*_8$ are apparently: $\{1,3,5,7\}$ - missing $2$, $4$ and $6$.
The only possible answer I can think of is that because <1>,<3>,<5> and <7> can generate the whole group - $2$, $4$ and $6$ are omitted from $\mathbb{Z}_8$ in the first place, meaning $*$ does just remove the zero as I thought.
Can someone explain this to me as the notes I'm using aren't helping!

Taking the elements coprime with n means those who have no common prime factor with n. This means we take only those who have an inverse (and are not zero divisors, which is saying the same thing dealing with finite groups). In the first example with $\mathbb{Z}/8$ , we keep the classes representatives as 1, 3, 5, 7 and make them into a abelian multiplicative group named $\mathbb{Z}^*_8$ . Some use an *, others like Andrew Baker from University of Glasgow use a x. He would write $(\mathbb{Z}/8)^×$. There is a group isomorphism with $\mathbb{Z}^*_8$ and  $\mathbb{Z}/2$ x $\mathbb{Z}/2$.
This shows the isomorphism:

http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf
