Chapter 4.4, Problem 3E

In Exercises 3 and 4, let $G$ be the octic group $D_4=\{e,\alpha ,{\alpha }^2,{\alpha }^3,\beta ,\gamma ,\Delta ,\theta \}$ in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section.

Let $H$ be the subgroup $\{e,\beta \}$ of the octic group $D_4$.

Find the distinct left cosets of $H$ in $D_4$, write out their elements, partition $D_4$ into left cosets of $H$, and give $[D_4:H]$.

Find the distinct right cosets of $H$ in $D_4$, write out their elements, and partition $D_4$ into right cosets of $H$.

Example 12

Using the notational convention described in the preceding paragraph, we shall write out the dihedral group $D_4$ of rigid motions of a square

The elements of the group $D_4$ are as follows:

1. the identity mapping $e=(1)$

2. the counterclockwise rotation $\alpha =(1,2,3,4)$ through ${90}^0$ about the center $O$

3. the counterclockwise rotation ${\alpha }^2=(1,3)(2,4)$ through ${180}^0$ about the center $O$

4. the counterclockwise rotation ${\alpha }^3=(1,4,3,2)$ through ${270}^0$ about the center $O$

5. the reflection $\beta =(1,4)(2,3)$ about the horizontal line $h$

6. the reflection $\gamma =(2,4)$ about the diagonal $d_1$

7. the reflection $\Delta =(1,2)(3,4)$ about the vertical line $v$

8. the reflection $\theta =(1,3)$ about the diagonal $d_2$.

The dihedral group $D_4=\{e,\alpha ,{\alpha }^2,{\alpha }^3,\beta ,\gamma ,\Delta ,\theta \}$ of rigid motions of the square is also known as the octic group. The multiplication table for $D_4$ is requested in Exercise 20 of this section.