- #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]=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?