Question

(a) If $\mathit{a}\mathbf{\in }\mathit{G}$and $\mathbf{\left|}\mathbf{a}\mathbf{\right|}\mathbf{=}\mathbf{12}$, find the orders of each of the elements $\mathit{a}\mathbf{,}{\mathit{a}}^{\mathbf{2}}\mathbf{,}{\mathit{a}}^{\mathbf{3}}\mathbf{,}\mathbf{\dots }\mathbf{,}{\mathit{a}}^{\mathbf{11}}$.

Short answer

$a^i$	$\left|a^i\right|$
a	12
$a^2$	6
$a^3$	4
$a^4$	3
$a^5$	12
$a^6$	2
$a^7$	12
$a^8$	3
$a^9$	4
$a^{10}$	6
$a^{11}$	12

Step-by-step solution

Determine given functions

Consider the given functions,

$a\in G$and$\left|a\right|=12$.

Determine the orders of elements

Now finding the orders of elements,

$a^i$	$\left|a^i\right|$
a	12
$a^2$	6
$a^3$	4
$a^4$	3
$a^5$	12
$a^6$	2
$a^7$	12
$a^8$	3
$a^9$	4
$a^{10}$	6
$a^{11}$	12