Question

What is the order of each group: (b).$D_4$

Short answer

The order of $D_4$ is $8$

Step-by-step solution

Order of a group

The order of group $G$ is defined as the number of elements in the group.

Definition of D4

It is defined as a symmetry of a regular polygon (here, square), whose center is fixed at origin by rotations and reflections.

Here, the rotations are given by:

$r_0,r_1,r_2,r_3$which are, $0{}^{\circ },90{}^{\circ },180{}^{\circ },270°$And, reflections are given by:

$d=$ reflection in X-axis

$t=$reflection in Y-axis

$h=$reflection in the linerole="math" localid="1653705807921" $y=x$

$v=$reflection in the line $y=-x$

Conclusion

Clearly, by the definition we have, $D_4=\left\{r_0,r_1,r_2,r_3,d,t,h,v\right\}$

So, the order of $D_4$ is $8$.