Normal subgroups of a non-albelian group

1. Dec 18, 2008

futurebird

I have the table for a non-albelian group. I know the subgroups of this group. I need to know which subgroups are normal. How can I tell?

2. Dec 19, 2008

latentcorpse

3. Dec 19, 2008

futurebird

So you need to check every single entry?

4. Dec 19, 2008

latentcorpse

Well some of these groups will be infinite so that's impossible but for finite groups I guess it would work but would be a tad tedious. We're looking for a more generic proof.

Consider the following,
Let $H \subset G$. The group $G$ is abelian and therefore has commutivity of elements by design i.e.
$ah=ha$
However, this holds $\forall h \in H$ and $\forall a \in G$
$\Rightarrow aH=Ha \Rightarrow a^{-1}aH=a^{-1}Ha$
$a^{-1}Ha=H$

5. Dec 19, 2008

futurebird

But my group isn't abelian.

6. Dec 19, 2008

latentcorpse

haha i am being silly...let me reconsider

7. Dec 19, 2008

latentcorpse

could u post the question?

8. Dec 19, 2008

futurebird

It's for a take-home final so I'm trying to ask for help on the concepts without doing that. I'll post it after I turn it in.