Subgroups of dihedral group and determining if normal

[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.

 

[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.

 

[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}

 

[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??

 

[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?

 

[micromass]

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?

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

 

[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.

 

[micromass]

Can you list all the elements in your group??

 

[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}

 

[micromass]

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}

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

 

[sleventh]

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

 

[lavinia]

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.

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.

 

[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.

 

[lavinia]

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.

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.

 

[sleventh]

Excellent. Thank you very much :)

 

[spamiam]

this is a very simple group.

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

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

 

[lavinia]

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

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

They are normal

 

[spamiam]

They are normal

Right, but you said the group was "very simple." <Cough, cough> http://en.wikipedia.org/wiki/Simple_group

I have a feeling jokes aren't as funny when you have to explain them...

 

[micromass]

Right, but you said the group was "very simple." <Cough, cough> http://en.wikipedia.org/wiki/Simple_group

I have a feeling jokes aren't as funny when you have to explain them...

That's so characteristic of you...

 

[sleventh]

oh my, that got me micromass haha :)