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The question comes from the problem that the $H$ is a maximal normal subgroup of $G$ $\Leftrightarrow$ $G/H$ is simple. The overall proof is mostly done, but I feel some difficulty to understand the why does H contain in $K$ if f($K$)=$N$ and $N$ is a subgroup of the quotient group $G/H$. I am going to use this result to generate the "if" part use contradiction. I have already proved the converse part. It might be weird to stuck at such a strange place. Thanks for your help.

Since I have only learnt the 1-3 isomorphism theorem, I really don't have any idea about the fourth one. But I hope by the primary knowledge I could solve the problem.

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    Try to prove that the subgroups contain a normal subgroup $H$ is in one to one correspondence with the subgroups of $G/H$. I assume you hypothesize $H\subset K\subset G$, and $f$ is the quotient map. Your question can be answered directly if you notice $H$ is normal in $K$ as well.2012-04-11
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    Please edit your question to indicate what $f$ stands for. Also please edit to clarify whether "why does $H$ contain in $K$" means "why does $H$ contain $K$" or "why is $H$ contained in $K$".2012-04-11

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