a). Find all the left and right cosets of $<τ>$ in $D_8$.
$<τ>$ is reflection and I know that there are two reflections about a diagonal. So I am wondering how to represent this idea as left and right cosets. My book did not go into detail about cosets or give any examples.
b). Find the index of $<ρ^2 τ>$ in $D_8$.
So from my understanding, $D_8$ is a group of order $8$ which is representative of a $4-gon$. So then there are $19$ total symmetries. There should be a total of two reflections about a single diagonal axis with rotation by $\pi$ from the first diagonal. My question for this is how do I find the index from this? My book did a poor job of defining index and cosets. I know I have to find the number of left cosets of $<ρ^2 τ>$ in $D_8$ but I don't know how to show this.
Any help would be appreciated.
Best Answer
Your "understanding" is way off!
"D8" is not "a 4-gon". D8 is a group with eight elements, one possible interpretation of which is the group of rotations and reflections of a square. There are not "19 elements" in the group, there are, as I said, 8. There are not "two reflections about an axis", there are two axes with one reflection about each. There are also two more reflections, about the lines through the center parallel to two of the sides.
Now, what are $\tau$ and $\rho$?