What Are Some Normal Subgroups of D4?

[kathrynag]

Homework Statement

(a) Show that if N and H are subgroups of G such that N is normal to G and N < H < G,
then N is normal to H.
(b) Find subgroups N and H of D4 such that N is normal H and H is normal to D4, but N is NOT a
normal subgroup of D4.
I

Homework Equations

The Attempt at a Solution

a) if xN =Nx for x in G then xN = Nx for x in H
I don't know where to go with this
b) I'm stuck on picking subgroups

 

[micromass]

Ok, so you've solved a.

Now, can you list me the subgroups in D4?

 

[kathrynag]

Well D4={e,r,r^2,r^3,f,fr, fr^2, fr^3}
I'm not really sure on all of the subgroups.
I have a couple:
{e,f}
{e,r,r^2,r^3}

 

[micromass]

Well, question b is actually more trial-and-error. So I can't give you more advice then: try to find every subgroup and start checking for normality.

There are however a few theorems you might consider, which may ease the task:
1) every subgroup of an abelian group is normal
2) every subgroup of index 2 is normal

 

[micromass]

Another thing that might help you: there are 10 subgroups of D4...

 

[kathrynag]

{e}
{e,r,r^2,r^3}
{e,r^2}
{e,f}
{e,rf}
{e,fr^2}
{e,fr^3}
{r^2,f}
{r^2,fr}
Ok, so that gives me 9

 

[Juanriq]

In order for it to be a subgroup, don't forget that it has to contain the identity.

 

[micromass]

{r^2,f}
{r^2,fr}

These two are not subgroups. They do not contain the identity. And moreover, the product of two elements is not necessairily in the group. However, they do generate subgroups. Which one?

The tenth subgroup is just D4 itself..

Now, once you found all the subgroups, which ones are normal?