Chapter 3.4, Problem 16E

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. (Sec. $3.5,\#3,6\ll$, Sec. $4.6,\#17\ll$).

Find an isomorphism from the additive group ${\mathbb{Z}}_4=\left\{{\left[0\right]}_4,{\left[1\right]}_4,{\left[2\right]}_4,{\left[3\right]}_4\right\}$ to the multiplicative group of units $U_5=\left\{{\left[1\right]}_5,{\left[2\right]}_5,{\left[3\right]}_5,{\left[4\right]}_5\right\}\subseteq {\mathbb{Z}}_5$.

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

Repeat Exercise $14$ where $G$ is the multiplicative group of units $U_{20}$ and $G^{\prime }$ is the cyclic group of order $4$. That is,

$G=\left\{\left[1\right],\left[3\right],\left[7\right],\left[9\right],\left[11\right],\left[13\right],\left[17\right],\left[19\right]\right\},$

$G^{\prime }=\langle a\rangle =\{e,a,a^2,a^3\}$

Define $\varnothing :G\to G^{\prime }$ by

$\varnothing (\left[1\right])=\varnothing (\left[11\right])=e$

$\varnothing (\left[3\right])=\varnothing (\left[13\right])=a$

$\varnothing (\left[9\right])=\varnothing (\left[19\right])=a^2$

$\varnothing (\left[7\right])=\varnothing (\left[17\right])=a^3$.