I could use some help on this proof:
Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove that H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.
Yes! I just deleted my link because it was a diaster as an answer. I was just being lazy. I type the proof up real soon right now. Be Patient.
I will use the following really useful theorem.
Theorem: Let G be a group and H be a subgroup of G. Then H is a normal subgroup of G if and only if ghg^{-1} in H for all g in G and h in H.
I will prove only one direction, the one I think is harder. And I leave the second direction which will be very similar as an excersice. I recommend to post your proof of the other direction for me to check it.
In the proof I shall use a triangle symbol. It means "normal subgroup of".
Thanks for the help. I think I've got the other direction down. Let me know if I did something wrong.
Claim: If H is a normal subgroup of G, then H/N is a normal subgroup of G/N.
Since N is a normal subgroup of G and H is a subset of G, we know hNh^(-1) is an element of N, which means H/N exists. Since H is a normal subgroup of G, ghg^(-1) is an element of H for all g in G. So, ghg^(-1)N = gN(hN)g^(-1)N is an element of H/N. Thus, H/N is a subgroup of G/N. Q.E.D.
Sorry for the lack of LaTeX, I still need to learn how to use it.
That is basically it. It is just stated in a very strange way.
Nothing wrong here. But if it was on exam I would state the initial hypothesis such as, N is normal subgroup of G and H is a subgroup of G.Claim: If H is a normal subgroup of G, then H/N is a normal subgroup of G/N.
No need to say "subset of" if it is a subgroup it is certainly a subset .Since N is a normal subgroup of G and H is a subset of G,
Perhaps you want to say "we know that hNh^(-1)=N for all h in H.we know hNh^(-1) is an element of N
You mean well-defined. No need to say that. We already know that N is normal subgroup of G so definitely H.which means H/N exists.
And for all h in H.Since H is a normal subgroup of G, ghg^(-1) is an element of H for all g in G.
That is the most important part of the proof. That is all good.So, ghg^(-1)N = gN(hN)g^(-1)N is an element of H/N. Thus, H/N is a subgroup of G/N. Q.E.D.
It is very readable. Group theory looks easy.Sorry for the lack of LaTeX, I still need to learn how to use it.