Chapter 3.4, Problem 11E

According to Exercise $33$ of section $3.1$, if $n$ is prime, the nonzero elements of ${\mathbb{Z}}_n$ form a group $U_n$ with respect to multiplication. For each of the following values of $n$, show that this group $U_n$ is cyclic. (Sec. $3.1,\#33\gg$)

a. $n=7$

b. $n=5$

c. $n=11$

d. $n=13$

e. $n=17$

f. $n=19$

a. Let $G=\left\{\left[a\right]|\left[a\right]\ne \left[0\right]\right\}\subseteq {\mathbb{Z}}_n$. Show that $G$ is a group with respect to multiplication in ${\mathbb{Z}}_n$ if and only if $n$ is a prime. State the order of $G$. This group is called the group of units in ${\mathbb{Z}}_n$ and designated by $U_n$.

b. Construct a multiplication table for the group $U_7$ of all nonzero elements in ${\mathbb{Z}}_7$, and identify the inverse of each element.