1. ## D8 subgroups

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

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

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

2. ## Re: D8 subgroups

See Deveno's posts in answer to my post "Normal Subgroups in D4 "

Peter

3. ## Re: D8 subgroups

Originally Posted by Bernhard
See Deveno's posts in answer to my post "Normal Subgroups in D4 "

Peter
So the Centralizer of each element is a subgroup then?

And normal subgroups are in the center?

4. ## Re: D8 subgroups

Sorry.

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

Peter

5. ## Re: D8 subgroups

Originally Posted by Bernhard
Sorry.

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

Peter
It didn't matter I learned something from it.

So the cyclic subgroups are $\displaystyle <r>=<r^3>$, correct?

6. ## Re: 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

7. ## Re: D8 subgroups

Originally Posted by Bernhard
Yes, but I think there may be others of order 2 such as <s> = {s, e} and <sr> = {sr, e}

Do you agree?

Peter
How were you able to identify those at cyclic groups?

8. ## Re: 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