Multiplication group for $\mathbb Z_n$ modulo $n$ By definition:
Let $\mathbb Z^+_n = \{[0],[1],[2],\ldots,[n−1]\}$
$\mathbb Z^+_4 = \{[0],[1],[2],[3]\},$
but
how $\mathbb Z^*_{12}$ is $\{[1],[5],[7],[11]\}$ ?
how $\mathbb Z^*_{7}$ is $\{[1],[2],[3],[4],[5],[6]\}$ ?
 A: You should distinguish between the additive groups $Z_4$ and
$Z_{12}$, which has $\{[0],[1],[2],[3],[4],...,[11]\}$,
and the multiplicative groups $Z_4^*$, which has $\{[1],[3]\}$, and $Z_{12}^*$, which has $\{[1],[5],[7],[11]\}$.
Those marked with $^*$ have the elements with multiplicative inverses.
Addendum to answer additional question in comment from OP:
The elements with multiplicative inverses in the ring $Z_{12}$ are those
whose representatives are relatively prime to $12$, i.e.,  not multiples of $2$ or $3$.
A: $\require{cancel}$
\begin{align}
\frac 1 {12} & = \frac 1 {12} & \text{No cancellation occurs here.} \\[6pt]
\frac 2 {12} & = \frac {\cancel2\times1}{\cancel2\times6} = \frac 1 6 \\[6pt]
\frac 3 {12} & = \frac {\cancel3\times 1}{\cancel3\times 4} = \frac 1 4 \\[6pt]
\frac 4 {12} & = \frac{\cancel4\times1}{\cancel4\times3} = \frac 13 \\[6pt]
\frac 5 {12} & = \frac 5 {12} & \text{No cancellation occurs here.} \\[6pt]
\frac 6 {12} & = \frac{\cancel6\times1}{\cancel6\times2} = \frac 1 2 \\[6pt]
\frac 7 {12} & = \frac 7 {12} & \text{No cancellation occurs here.} \\[6pt]
\frac 8 {12} & = \frac{\cancel4\times2}{\cancel4\times3} = \frac 2 3 \\[6pt]
\frac 9 {12} & = \frac{\cancel3\times3}{\cancel3\times4} = \frac 3 4 \\[6pt]
\frac{10}{12} & = \frac{\cancel2\times5}{\cancel2\times6} = \frac 5 6 \\[6pt]
\frac{11}{12} & = \frac{11}{12} & \text{No cancellation occurs here.} \\ {}
\end{align}
A: For any $n$ (positive integer), those elements of $G = \mathbf{Z}/n\mathbf{Z}$ whose representatives are realatively prime to $n$ make up the multiplicative group of units of $G$, denoted $G^\times$. You can compare this with your example $n = 12$, which are, as J.W. Tanner wrote, not multiples of $2$ or $3$.
The additive group $G$ gets its additive (abelian) structure from the rules for modular arithmetic (if two sums are congruent modulo $n$, then they are equal in $G$, and similarly for products).
