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The Trouble with Normal Subgroups

  • #1
Tom Mattson
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[SOLVED] The Trouble with Normal Subgroups

Homework Statement


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


Homework Equations


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, [itex][G:K] =2[/itex] and [itex][K:H]=2[/itex]). Finding such groups is fairly easy, but finding them with [itex]H[/itex] not normal in [itex]G[/itex] has proved elusive.

Here are my false starts.

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

Result: No good, [itex]H[/itex] is normal in [itex]G[/itex]

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

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.

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

Once again, [itex]H[/itex] is normal in both [itex]K[/itex] and [itex]G[/itex].

I also did some goofing around with [itex]S_n[/itex] and [itex]A_n[/itex], but to no avail.

Little help?
 

Answers and Replies

  • #2
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Try the dihedral group with order 8.
 
  • #3
Tom Mattson
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That was my first attempt. It didn't work.
 
  • #4
Tom Mattson
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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, [itex]\left<ab^2\right>[/itex] is normal in [itex]D_8[/itex], [itex]\left<a,b^2\right>[/itex] is normal in [itex]\left<ab^2\right>[/itex], but [itex]\left<ab^2\right>[/itex] is not normal in [itex]D_8[/itex], as [itex]b^{-1}ab^2b=abb^3=ab^4=a[/itex], which is not in [itex]\left<ab^2\right>[/itex].

So it was [itex]D_8[/itex] all along...
 

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