The Trouble with Normal Subgroups

[quantumdude]

[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.

Homework Equations

None.

The Attempt at a Solution

I have many attempts.

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<img src="/styles/physicsforums/xenforo/smilies/arghh.png" class="smilie" loading="lazy" alt=":H" title="Gah! :H" data-shortname=":H" />]=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=&lt;b&gt;, the subgroup of order 4 generated by b
H=&lt;b^2&gt;, 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&lt;b\right&gt;
H=\left&lt;b^2\right&gt;

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?

 

[zhentil]

Try the dihedral group with order 8.

 

[quantumdude]

That was my first attempt. It didn't work.

 

[quantumdude]

Oh brother! I've got it. Like a dummy I was fixated on cyclic subgroups. I just considered a subgroup with two generators, and I got it.

FYI, \left&lt;ab^2\right&gt; is normal in D_8, \left&lt;a,b^2\right&gt; is normal in \left&lt;ab^2\right&gt;, but \left&lt;ab^2\right&gt; is not normal in D_8, as b^{-1}ab^2b=abb^3=ab^4=a, which is not in \left&lt;ab^2\right&gt;.

So it was D_8 all along...