Question

(a) If $\mathit{a}\mathbf{\in }\mathit{G}$ and${\mathit{a}}^{\mathbf{12}}\mathbf{=}\mathit{e}$ , what order can$\mathit{a}$ possibly have?

Short answer

For$k=1$ and$a=0$

For$k=2$ androle="math" localid="1654255297629" $a=6$

For $k=3$and$a=4$

For $k=4$and$a=3$

For $k=6$and $a=2$

For $k=12$and$a=1$

Step-by-step solution

Determine order of a

Consider the given functions, $a\in G$ and $a^{12}=e$

Let’s assume that, the order of role="math" localid="1654255469356" $a$is denoted by $k$.

From theorem 7.9 (a), $a^{12}=e$and it follows that,

$k|12$

Thus,$k=\{1,2,3,4,6,12\}$

Determine order of (Further simplification)

The value of $k$listed above. There are examples of group of element $a$. And $a$is order of $k$.

Now taking a group ${\mathbb{Z}}_{12}$

For$k=1$ and$a=0$

For $k=2$androle="math" localid="1654255789159" $a=6$

For $k=3$and$a=4$

For $k=4$and$a=3$

For $k=6$and $a=2$

For $k=12$and$a=1$