Proving Core of a Group is Normal in G and Subset of H

[Punkyc7]

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


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

 

[Office_Shredder]

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)

 

[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.

 

[Office_Shredder]

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

 

[Punkyc7]

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