I'm looking at the proof of the third isomorphism theorem and I have a few questions.
The statement of the theorem is as follows: Let K be a normal subgroup in G, let H be a subgroup of G containing K.
1) H normal in G iff H/K is normal in G/K
2) G/H is isomorphic to G/K / H/K, for some H normal in G.
The proof of 1) is as follows:
Assume H is normal in G. Then for h in H, g in G, $(gK)(hK){(gK)^{-1}}$ = $ghg^{-1}K$ is in H/K, since $ghg^{-1}$ is in H. Why does $ghg^{-1}$ in H imply that $ghg^{-1}K$ is in H/K?
The proof continues to say that H/K is normal in G/K. How does this follow?
Thanks for your help.
EDIT: Oh I think I understand it now. H/K is cosets of K in H, so since ${ghg^{-1}}$ is in H, $ghg^{-1}K$ is in H/K. In addition, I am trying to show that $(gK)(hK){(gK)^{-1}}$ is in H/K, since that will show H/K is normal in G/K. Is this correct...?
For the first part, you are correct. The elements of $H/K$ are the cosets of $K$ in $H$, which are all of the form $hK$ for $h\in H$. In your particular case, since $H$ is normal in $G$, you get $ghg^{-1}\in H$, so that $(ghg^{-1})K$ is a left coset.
For the second part, to see that $H/K$ is normal in $G/K$, you need to show that if $gK$ is any left coset of $K$ in $G$ (i.e., any element of $G/K$), and $hK$ is any left coset of $K$ in $H$ (i.e., any element of $H/K$), then $(gK)hK(gK)^{-1}\in H/K$. So choose particular elements of those cosets, say $gk_1$ and $hk_2$; you want to show that $gk_1hk_2(gk_1)^{-1} = h'k'$ for some $h'\in H$ and $k'\in K$. To see this, you will need to use the fact that $K$ is normal in $G$ and that $H$ is normal in $G$, which allows you to say things like $gk = k_1g$ and $gh = h_1g$.