Question

10. Find the order of every element in each group.

(d)$D_4$

Short answer

The order of every element in the group$D_4$ is $\left\{1,4,2,4,2,2,2,2\right\}$.

Step-by-step solution

Order of an element

The order of an element x in a group G is the smallest positive integer n such that $x^n=e$. The order of the element x is denoted by $\left|x\right|$.

Order of elements in D4

The number of elements in the group$D_4$is 8 which are $\left\{r_0,r_1,r_2,r_3,d,h,t,v\right\}$.

By the definition the order of identity element$r_0$is 1, the order of elements$\left\{r_2,d,h,t,v\right\}$is 2 and order of$\left\{r_1,r_3\right\}$is 4.

The order of each element in $D_4$is given in the below table:

Element	Order
$r_0$	1
$r_1$	4
$r_2$	2
$r_3$	4
d	2
h	2
t	2
v	2

Therefore, the order of every element in the group$D_4$ is $\left\{1,4,2,4,2,2,2,2\right\}$.