Chapter 3.1, Problem 33E

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 $U_n$. (Sec $3.3\#5$, Sec $3.4\#11$, Sec $3.5\#19$)

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. (Sec $4.4,\#1,19,26$)

Sec $3.3\#5\ll$

5. Exercise $33$ of section $3.1$ shows that $U_{13}\subseteq {\mathbb{Z}}_{13}$ is a group under multiplication.

List the elements of the subgroup $\langle \left[4\right]\rangle$ of $U_{13}$, and state its order.

List the elements of the subgroup $\langle \left[8\right]\rangle$ of $U_{13}$, and state its order.

Sec $3.4\#11\ll$

11. If $n$ is a 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.

$n=7$

b. $n=5$

c. $n=11$

d. $n=13$

e. $n=17$

f. $n=19$

Sec $3.5\#19\ll$

19. If $n$ is a prime, $U_n$, the set of nonzero elements of ${\mathbb{Z}}_n$, forms a group with respect to multiplication. Prove or disprove that the mapping $\varnothing :U_n\to U_n$ defined by the rule in Exercise $18$ is an automorphism of $U_n$.

Construct a multiplication table for the group $U_7$ of all nonzero elements in ${\mathbb{Z}}_7$, and identify the inverse of each element. (Sec $4.4,\#1,19,26$)

Sec $4.4,\#1\ll$

1. Consider $U_{13}$, the groups of units in ${\mathbb{Z}}_{13}$ under multiplication. For each of the following subgroups $H$ in $U_{13}$, partition $U_{13}$ into left cosets of $H$, and state the index $\left[U_{13}:H\right]$ of $H$ in $U_{13}$

$H=\langle \left[4\right]\rangle$

b. $H=\langle \left[8\right]\rangle$

Sec $4.4,\#19\ll$

19. Find the order of each of the following elements in the multiplicative group of units $U_p$.

$\left[2\right]$ for $p=13$

b. $\left[5\right]$ for $p=13$

c. $\left[3\right]$ for $p=17$

d. $\left[8\right]$ for $p=17$

Sec $4.4,\#26\ll$

26. Let $p$ be prime and $G$ the multiplicative group of units $U_p=\left\{\left[a\right]\in {\mathbb{Z}}_p|\left[a\right]\ne \left[0\right]\right\}$. Use Langrange’s Theorem in $G$ to prove Fermat’s Little Theorem in the form ${\left[a\right]}^p=\left[a\right]$ for any $a\in \mathbb{Z}$.