# Derived subgroup proofs

Let G' be the derived subgroup. Prove:

a) If G' is a subgroup of H and H is a subgroup of G, show that H is normal in G;

b) If K is a normal subgroup of G, then K' is a normal subgroup of G

c) Suppose f:G --> H is an epimorphism, with ker(f)=K, then H is abelian iff G' is a subgroup of K.

I need a detailed proof of this to study please.

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#### Solution Preview

(a) Since H contains G', it contains all commutators of the form ghg^{-1} h^{-1}, where g is in G, and h is in H, hence

ghg^{-1} belongs to H, i.e. H is normal in G.

(b) Suppose x, y are in K; then g[x, y]g^{-1} = [gxg^{-1}, gyg^{-1}] belongs to K, since K is normal in G.

Since K' is generated by the commutators ...

#### Solution Summary

This provides examples of several proofs regarding a derived subgroup.

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