Fundamental theorem of galois theory

[Pratibha]

In fundamental theorem of galois theory,(statement): given that K be galois extension of F, G(K/F) be its galois group, S(K) be the set of all subfields of K containing F & S(G) be the set of all subgroups of G(K/F),
mapping g: from S(K) to S(G) defined by g(H)=G(K/H),
mapping h: from S(G) to S(K) defined by h(L)=H
then prove:
(i)composite map goh & hog are identity maps.
(ii)if H\in S(K) then m(H)=H for all m in G(K/F) iff H/F is normal.
(iii)H/F is normal iff G(K/H) is normal subgroup of G(K/F).
(iv) if H/F is normal then G(K/F)/G(K/H) is isomorphic to G(H/F).
my question is that
in (iv) part while proving G(K/F)/G(K/H) is isomorphic to G(H/F), we define a mapping f:G(K/F) to G(H/F) and we prove f is onto homomorphism(so that we can apply fundamental theorem of homomorphism), how f is onto homomorphism?

 

[jvicens]

To prove that f is an onto homomorphism, we need to show that it satisfies two conditions:

1. f is a homomorphism: This means that for any two elements x and y in G(K/F), f(xy) = f(x)f(y). This can be shown by first noting that G(K/F) is a group under multiplication, and then using the definition of f to show that it satisfies the homomorphism property.

2. f is onto: This means that for every element z in G(H/F), there exists an element x in G(K/F) such that f(x) = z. To show this, we can use the fact that H/F is a normal subgroup of G(K/F), which means that for any element z in G(H/F), there exists an element x in G(K/F) such that xH = zH. This implies that f(x) = z, and hence f is onto.

Therefore, by showing that f is a homomorphism and onto, we can conclude that it is an onto homomorphism, and thus we can apply the fundamental theorem of homomorphism to prove that G(K/F)/G(K/H) is isomorphic to G(H/F).