# The Trouble with Normal Subgroups

Staff Emeritus
Gold Member
[SOLVED] The Trouble with Normal Subgroups

## Homework Statement

Find an example of a group $G$ and subgroups $H$ and $K$ such that $H$ is normal in $K$, $K$ is normal in $G$, but $H$ is not normal in $G$.

None.

## The Attempt at a Solution

I have many attempts. :grumpy:

In the beginning I was fixated on groups of orders 8 and 16, because I figured it would be easy to find nested subgroups with index 2 (that is, $[G:K] =2$ and $[K ]=2$). Finding such groups is fairly easy, but finding them with $H$ not normal in $G$ has proved elusive.

Here are my false starts.

Attempt 1:
$G=D_8$, the dihedral group of order 8
$K=<b>$, the subgroup of order 4 generated by $b$
$H=<b^2>$, the subgroup of order 2 generated by $b^2$

Result: No good, $H$ is normal in $G$

Attempt 2:
$G=Q_8$, the quaternionic group of order 8
$K=\left<b\right>$
$H=\left<b^2\right>$

I didn't even bother finishing this one. It became obvious that this attempt would fail for the same reason that the last one failed.

Attempt 3:
Here's where I tried to be more clever.

$G=GL_2\left(\mathbb{Z}_3\right)$
$K=SL_2\left(\mathbb{Z}_3\right)$
$H=$ the subgroup generated by the elements of order 4 in $K$ (this is a subgroup of order 8).

Once again, $H$ is normal in both $K$ and $G$.

I also did some goofing around with $S_n$ and $A_n$, but to no avail.

Little help?

Related Calculus and Beyond Homework Help News on Phys.org
Try the dihedral group with order 8.

Staff Emeritus
Gold Member
That was my first attempt. It didn't work.

Staff Emeritus
FYI, $\left<ab^2\right>$ is normal in $D_8$, $\left<a,b^2\right>$ is normal in $\left<ab^2\right>$, but $\left<ab^2\right>$ is not normal in $D_8$, as $b^{-1}ab^2b=abb^3=ab^4=a$, which is not in $\left<ab^2\right>$.
So it was $D_8$ all along...