Chapter 3.5, Problem 6E

Find an isomorphism from the additive group ${\mathbb{Z}}_6=\left\{{\left[a\right]}_6\right\}$ to the multiplicative group of units $U_7=\{{\left[a\right]}_7\in {\mathbb{Z}}_7|{\left[a\right]}_7\ne {\left[0\right]}_7\}$.

Sec. $3.4,\#16\gg$

For an integer $n>1$, let $G=U_n$, the group of units in ${\mathbb{Z}}_n$ –that is, the set of all $\left[a\right]$ in ${\mathbb{Z}}_n$ that have multiplicative inverses. Prove that $U_n$ is a group with respect to multiplication.