1. ## Dihedral Group

I am trying to work out all the subgroups of $\displaystyle D_8$, the dihedral group, i.e. the symmetry group of the square.

I am not sure how to do this. I know of course they must have order 1,2,4,8, so the only real problem is working out the subgroups of order 4. Is there a quick way to do this? Thanks for any help with this

2. Originally Posted by slevvio
I am trying to work out all the subgroups of $\displaystyle D_8$, the dihedral group, i.e. the symmetry group of the square.

I am not sure how to do this. I know of course they must have order 1,2,4,8, so the only real problem is working out the subgroups of order 4. Is there a quick way to do this? Thanks for any help with this

$\displaystyle D_8=\{s,t\,;\,s^2=t^4=1\,,\,sts=t^3\}$ , so now write down explicitly the elements of this group and their orders, and solve.

Tonio

3. Thanks but I don't quite understand that notation, what does the ' s,t ' mean?

4. Originally Posted by slevvio
Thanks but I don't quite understand that notation, what does the ' s,t ' mean?

You're studying dihedral groups and haven't seen the above? You can think of t as a rotation in 90 degrees, and s is a reflection.

Tonio

5. Never seen that notation for a group before although I am aware of the rotations and reflections. Thanks for clarifying.

I know there is an obvious subgroup of order 4 where you take the subgroup of rotations, but how can I work out there are only TWO other subgroups of order 4 and what they are exactly?

6. Originally Posted by slevvio
Never seen that notation for a group before although I am aware of the rotations and reflections. Thanks for clarifying.

I know there is an obvious subgroup of order 4 where you take the subgroup of rotations, but how can I work out there are only TWO other subgroups of order 4 and what they are exactly?

If you don't write explicitely the group's elements, better with my notation or any other way, I can't see how you'll succeed...sorry.

Tonio

7. Originally Posted by slevvio
Never seen that notation for a group before although I am aware of the rotations and reflections. Thanks for clarifying.

I know there is an obvious subgroup of order 4 where you take the subgroup of rotations, but how can I work out there are only TWO other subgroups of order 4 and what they are exactly?
You've never seen a group written out in terms of its elements?

How do you usually think of groups? For some groups such as this one, you can get some stuff done by thinking abstractly/geometrically about the group. But how can you REALLY work with a group without being able to write down what is IN the group?