Chapter 3.3, Problem 5E

Exercise $33$ of section $3.1$

shows that $U_{13}\subseteq Z_{13}$ is a group under multiplication.

a. List the elements of the subgroup $\langle \left[4\right]\rangle$

of $U_{13}$, and state its order.

b. List the elements of the subgroup $\langle \left[8\right]\rangle$

of $U_{13}$, and state its order.

Exercise 33 of section 3.1.

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 is 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.