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.