# 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.