Proving Normality of a Quotient Group: A Shortcut Method

[Hjensen]

I have a question I need to resolve before my exam on thursday. It relates to the following result:

Let $N$ be a normal subgroup of $G$, and let $K$ be any subgroup of $G$ containing $N$. Then $K/N$ is a subgroup of $G/N$. Furthermore, $K/N$ is normal in $G/N$ if and only if $K$ is normal in $G$.

The first part is rather simple, and half of the other statement is just from the third isomorphism theorem. What I want to prove is, that $K/N$ normal in $G/N$ implies that $K$ is normal in $G$. I suppose I could define a homomorphism like

$G\rightarrow G/N\rightarrow (G/N)/(K/N)$

with kernel $K$. That just seems like a lot of work to prove something which is probably rather simple. If I have to go through this at my exam, I'd prefer not to spend much time on this particular result. Does anyone have an idea for a short proof?

 

[micromass]

You need to prove for every $k\in K$ that $gkg^{-1}\in K$.
But if we do the calculation in G/N, then we get by normality of K/N that

$$[gkg^{-1}]=[g][k][g]^{-1}\in K/N$$

By definition, this means that there is a k' in K and a n in N such that $gkg^{-1}=k^\prime n$. But N is a subset of K, thus $gkg^{-1}\in K$.

 

[Zorba]

Using the isomorphism theorems here seems like "killing a fly with a nuke" or whatever the saying is (although your idea of that homomorphism does seem interesting.) micromass seems to have covered the rest.