Hw5 Problem 6

Problem 6

(a) Create a table for the group $\mathbb Z_8$ under addition.
(b) Compute the subgroups $\langle 0 \rangle$, $\langle 1 \rangle$, $\langle 2 \rangle$, $\langle 3 \rangle$, $\langle 4 \rangle$, $\langle 5 \rangle$, $\langle 6 \rangle$, $\langle 7 \rangle$.
(c) Which elements are generators for the group $\mathbb Z_8$?
(d) Give the subgroup diagram for the part (b) subgroups of $\mathbb Z_8$.

Solution
(a) Create a table for the group $\mathbb Z_8$ under addition.

(1)
\begin{array} {c||c|c|c|c|c|c|c|c} * & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 0 \\ \hline 2 & 2 & 3 & 4 & 5 & 6 & 7 & 0 & 1 \\ \hline 3 & 3 & 4 & 5 & 6 & 7 & 0 & 1 & 2 \\ \hline 4 & 4 & 5 & 6 & 7 & 0 & 1 & 2 & 3 \\ \hline 5 & 5 & 6 & 7 & 0 & 1 & 2 & 3 & 4 \\ \hline 6 & 6 & 7 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 7 & 7 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \end{array}

(b) Compute the subgroups $\langle 0 \rangle$, $\langle 1 \rangle$, $\langle 2 \rangle$, $\langle 3 \rangle$, $\langle 4 \rangle$, $\langle 5 \rangle$, $\langle 6 \rangle$, $\langle 7 \rangle$.

$\langle 0 \rangle = \{0\}$
$\langle 1 \rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$
$\langle 2 \rangle = \{0, 2, 4, 6\}$
$\langle 3 \rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$
$\langle 4 \rangle = \{0, 4\}$
$\langle 5 \rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$
$\langle 6 \rangle = \{0, 2, 4, 6\}$
$\langle 7 \rangle = \{0, 1, 2, 3, 4, 5, 6, 7\}$

(c) Which elements are generators for the group $\mathbb Z_8$?

The generators of the group $\mathbb Z_8$ are $1, 3, 5, 7$

(d) Give the subgroup diagram for the part (b) subgroups of $\mathbb Z_8$.

(2)
\begin{align} \mathbb Z_8 = \langle 1 \rangle = \langle 3 \rangle = \langle 5 \rangle = \langle 7 \rangle \end{align}
(3)
$$|$$
(4)
\begin{align} \langle 2 \rangle = \langle 6 \rangle \end{align}
(5)
$$|$$
(6)
\begin{align} \langle 4 \rangle \end{align}
(7)
$$|$$
(8)
\begin{align} \langle 0 \rangle \end{align}