# Quotienting out by M

## Main Question or Discussion Point

If ##N \trianglelefteq H ##, ##N \trianglelefteq G ##, and ##H \le G##, then is it true that ##H/N \le G/N##?

I want to use the result for a proof I am currently doing, but I am not sure it is true.
Is it enough just to note that if ##h_1,h_2\in H##, then ##(h_1N)(h_2^{-1}N) = h_2h_2^{-1}N \in H/N##?

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Mentor
If ##N \trianglelefteq H ##, ##N \trianglelefteq G ##, and ##H \le G##, then is it true that ##H/N \le G/N##?

I want to use the result for a proof I am currently doing, but I am not sure it is true.
Is it enough just to note that if ##h_1,h_2\in H##, then ##(h_1N)(h_2^{-1}N) = h_2h_2^{-1}N \in H/N##?
No, it's wrong. The index before last has to be ##1##

The rest is a yes.
E.g.: https://en.wikipedia.org/wiki/Isomorphism_theorems#Third_isomorphism_theorem

Mr Davis 97