Explore BrainMass
Share

Explore BrainMass

    Derived subgroup proofs

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com October 9, 2019, 10:39 pm ad1c9bdddf
    https://brainmass.com/math/discrete-math/derived-subgroup-proofs-227726

    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.

    $2.19