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!

This is a notational device. The $^*$ is being used to show that the group operation is multiplication, and the elements of the group are the elements of $\mathbb Z_8$ which are coprime to $8$. The identity is $1$.
The integers taken modulo $n$ inherit both addition and multiplication from $\mathbb Z$. If you take the elements coprime to $n$ you get a multiplicative group of order $\varphi (n)$ whose elements satisfy $$x^{\varphi(n)}=1$$This is the Euler-Fermat theorem, a generalisation of Fermat's Little Theorem.
