My modern algebra class is currently working on cosets and the theorem of Lagrange. I understand the concept of cosets, but the calculations of double cosets is completely stumping me. I could really use help working on this question.
Find all $(H,K)$-double cosets in $G$ where $G=D_{16}$, $H=\{1,r^4,sr,sr^5\}$, and $K=\{1,s\}$
I have that $D_{16}=\{1,r,r^2,\dotsc,r^7,s,sr,sr^2,\dotsc ,sr^7\}$ but don't know what to do from here.
You want to work as you do when you calculate cosets! Because double cosets partion the group you have to calculate $HxK$ until you ran out of $x$ (which is guaranteed since you have a finite). I will calculate one example, namely $r^2$:
$Hr^2K=(Hr^2)K=\{r^2,r^6,sr^3,sr^7\}K=\{r^2,r^6,sr^3,sr^7,r^2s,r^6s,sr^3s,sr^7s\}$.
I leave it tou you to make simplify the exprasions using the realations of the Dihedral group. Think you can continue from here?