Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H.

Question

Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H. Show that if K is a normal subgroup of G, then K is a normal subgroup of H.

Ive tried writing out what is implied from all of the givens, but i guess I'm just having trouble seeing a connection. K is a normal subgroup of G tells us that
aK=Ka for all a in G
therefore aKa^(-1)=K for all a in G
We want to show that aKa^(-1)=K for all a in G
I also have all of the implications of subgroups from my class, but i can't find one that appears to work.

Answer 1

Hint:

He who can do the more can do the less.

Answer 2

Hint: Taking $K$ as the kernel of some homomorphism $H \to P$ (which one?) for some group $P$ is another way to do this.

I think it's a little cleaner than the bare-handed way.