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## Main Question or Discussion Point

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

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

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 [itex]K/N[/itex] normal in [itex]G/N[/itex] implies that [itex]K[/itex] is normal in [itex]G[/itex]. I suppose I could define a homomorphism like

[itex]G\rightarrow G/N\rightarrow (G/N)/(K/N)[/itex]

with kernel [itex]K[/itex]. 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?

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

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 [itex]K/N[/itex] normal in [itex]G/N[/itex] implies that [itex]K[/itex] is normal in [itex]G[/itex]. I suppose I could define a homomorphism like

[itex]G\rightarrow G/N\rightarrow (G/N)/(K/N)[/itex]

with kernel [itex]K[/itex]. 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?