Easy question. How do I find the subgroups of $\displaystyle D_8$?

$\displaystyle D_8=<r,s:r^4=s^2=1, \ rs=sr^{-1}>$

$\displaystyle D_8=\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$

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- Oct 8th 2011, 04:12 PMdwsmithD8 subgroups
Easy question. How do I find the subgroups of $\displaystyle D_8$?

$\displaystyle D_8=<r,s:r^4=s^2=1, \ rs=sr^{-1}>$

$\displaystyle D_8=\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$ - Oct 8th 2011, 04:42 PMBernhardRe: D8 subgroups
See Deveno's posts in answer to my post "Normal Subgroups in D4 "

Peter - Oct 8th 2011, 04:52 PMdwsmithRe: D8 subgroups
- Oct 8th 2011, 04:57 PMBernhardRe: D8 subgroups
Sorry.

I was too quick in referrring you to my post - I was looking for**normal**subgroups only.

Peter - Oct 8th 2011, 05:07 PMdwsmithRe: D8 subgroups
- Oct 8th 2011, 06:03 PMBernhardRe: D8 subgroups
Yes, but I think there may be others of order 2 such as <s> = {s, e} and <sr> = {sr, e}

Do you agree?

Peter - Oct 8th 2011, 06:13 PMdwsmithRe: D8 subgroups
- Oct 8th 2011, 06:36 PMBernhardRe: D8 subgroups
Just by checking the subgroups generated by the elements concerned - no smart method - just working through the elements of the group checking the subgroups generated by each element. If the group concerned was very much bigger you would have to be more analytic I guess.

Another possibility is <$\displaystyle r^2 $> = {e, $\displaystyle r^2 $} and of course there is {e}.

Peter