# Subgroups of dihedral group and determining if normal

1. Dec 7, 2011

### sleventh

To find all subgroups you use the fact that by Legrange theorem and subgroup will divide the order of the group, so for the dihedral group D4 our subgroups are of order 1,2, and 4. I am unsure how to tell whether or not these groups will be normal or not.

2. Dec 7, 2011

### micromass

Whether a group will be normal or not does not (in general) depend on the order of the group. So only knowing the order will not suffice to determine normality or not. (except in some cases: subgroups of index 2 are always normal for example)

You'll need to list the explicit subgroups.

3. Dec 7, 2011

### sleventh

(with r being rotation and s being reflection) would the subgroup of order 1 be {r^4} and {s^2}, of order 2 be {r^2} and {s}, and the order 4 be {r}

4. Dec 7, 2011

### micromass

Yes, but there are more subgroups out there. You're missing some subgroups of order 2 and 4.

Can you also deduce which one you list is normal??

5. Dec 7, 2011

### sleventh

i am not quite sure, but my best guess would be r^2 because by any reflection of number if rotations you will be able to return to r^2, for the same reason r, and r^4. Is this correct? also, why is it that index 2 groups are normal?

6. Dec 7, 2011

### micromass

That is correct. But that aren't all the normal subgroups yet.

7. Dec 7, 2011

### sleventh

s^2 because it's the identity. I'm hesitant to say s because if you perform one reflection on s, so rs = s' then the reflections of s' will not be able to return to s by reflection.

8. Dec 7, 2011

### micromass

Can you list all the elements in your group??

9. Dec 7, 2011

### sleventh

there must be 8 because we have four sides and four rations, by each rotation acts on 2 sides. { r, r^2, r^3, r^4, s, s^2, rs, r^2s}

10. Dec 7, 2011

### micromass

s2 and r4 are equal. So you're missing one.

11. Dec 7, 2011

### sleventh

ah, right. Is the last r^3s?

12. Dec 7, 2011

### lavinia

this is a very simple group. I would just take a representation of it and play.

Think of the group as having two generators, a 90 degree rotation of the plane and a reflection around the y axis.

13. Dec 7, 2011

### sleventh

right, this is why I have been using the r, s notation. But I am still unsure how to tell if a subgroup is normal.

14. Dec 7, 2011

### lavinia

A subgroup is normal if all of its conjugates are in the group. If the group is cyclic yhen you only need to check this on a generator.

So for instance if you conjugate the 90 rotation by the reflection around the y axis you get its cube, a 270 degree rotation. So this subgroup is normal.

15. Dec 7, 2011

### sleventh

Excellent. Thank you very much :)

16. Dec 7, 2011

### spamiam

No it isn't, he just found a normal subgroup! :tongue:

I hope puns aren't ban-worthy...

17. Dec 8, 2011

### lavinia

They are normal

18. Dec 8, 2011

### spamiam

19. Dec 8, 2011

### micromass

That's so characteristic of you...

20. Dec 12, 2011

### sleventh

oh my, that got me micromass haha :)