# Core of a Group

1. Feb 26, 2012

### Punkyc7

Let H be a subgroup of G and define the core of H as such
core H={g$\in$G| g$\in$aHa^-1 for all a$\in$G}= $\bigcap${aHa^-1|a$\in$G}
Prove that the core of H is normal in G and core H$\subset$H.

I am having a hard time proving this because isn't the definition of core H basically saying the the core is normal?

2. Feb 26, 2012

### Office_Shredder

Staff Emeritus
It is pretty close to tautological, but as I like to say if it's so obvious it should be easy to prove. You need to do two things:

1) Prove core(H) is actually a subgroup
2) Prove that for a∈G, acore(H)a-1=core(H)

3. Feb 26, 2012

### Punkyc7

Ok so would I say something like
If e is in core H becaus aea^-1=e
Let g and h^-1 be in the core then

ag^a-1(aha^-1)=agh^-1a^-1 ..........is this right?
so its a a sub group by the one step subgroup test.

4. Feb 26, 2012

### Office_Shredder

Staff Emeritus
As long as you state that that's true for all a, that looks good to me

5. Feb 26, 2012

### Punkyc7

Ok thanks, I thought there might have been something I was missing