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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.

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  • $\begingroup$ Given $x\in G,$ and $H,K$ subgroups, the double coset is $HxK=\{hxk\mid h\in H,~k\in K\}.$ $\endgroup$ Jan 19 '18 at 21:14
  • $\begingroup$ @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. :) $\endgroup$ Jan 19 '18 at 23:40
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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?

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  • $\begingroup$ so would HrK=(Hr)K={r, r^5, rsr, rsr^5}K={r, r^5, rsr, rsr^5, rs, r^5s, rsrs, rsr^5s} $\endgroup$ Jan 19 '18 at 22:20
  • $\begingroup$ 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? $\endgroup$
    – Nick A.
    Jan 19 '18 at 22:39
  • $\begingroup$ Okay that is exactly what I was wondering, thanks. $\endgroup$ Jan 19 '18 at 22:43
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    $\begingroup$ Generaly in the site, if you find an answer thankful you also upvote (whether you asked the question or not) and if you are indeed the asker you can choose one of the answers as the best (which also awards you some reputation) :) $\endgroup$
    – Nick A.
    Jan 19 '18 at 22:49

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