The group $\displaystyle D_8$ has two generators $\displaystyle x, y$ with the relations $\displaystyle x^4 = e, y^2 = e, xy = yx^3$. Let $\displaystyle H$ be the cyclic group generated by $\displaystyle x$. Write down the four elements in $\displaystyle H$. Write down the four elements in the left coset $\displaystyle yH$ and the four elements in the right coset $\displaystyle Hy$. Show that $\displaystyle yH = Hy$.

This is what I have so far.

$\displaystyle H = <x> = \{e, x, x^2, x^3\}$

$\displaystyle yH = \{ye, yx, yx^2, yx^3\} = \{y, yx, yx^2, yx^3\}$

$\displaystyle Hy = \{ey, xy, x^2y, x^3y\} = \{y, x^2y, x^3y, yx^3\}$ (because $\displaystyle xy = yx^3$)

Now I am supposed to show that $\displaystyle yH = Hy$ and I cannot seem to manipulate the sets so they are equal. Thanks in advance for any help.