# Determining double cosets

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.

• Given $x\in G,$ and $H,K$ subgroups, the double coset is $HxK=\{hxk\mid h\in H,~k\in K\}.$ Jan 19 '18 at 21:14
• @AmberGladman, I edited your title to reflect your actual question. The previous was a bit vague. If you don't like the edit, I won't be "hurt" if you change it back. :) Jan 19 '18 at 23:40

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 $r^2$:
$Hr^2K=(Hr^2)K=\{r^2,r^6,sr^3,sr^7\}K=\{r^2,r^6,sr^3,sr^7,r^2s,r^6s,sr^3s,sr^7s\}$.
• Not really, $Hr=\{r,r^5,sr^2,sr^6\}$, according atleast to what you defined $H$ to be in the question. maybe there is something wrong there? Jan 19 '18 at 22:39