Please answer C and D!!!! 6.6. Let G be a group, let H C G and K C G be subgroups, and assume that K is a normalsubgroup of G.(a) Prove that HK {hk : h € H, kЄ K} is a subgroup of G.(b) Prove that H ŉ K is a normal subgroup of H and that K is a normal subgroup of HK.(c) Prove that HK/K is isomorphic to H/(HnK). (Hint. What is the kernel of the surjectivehomomorphism H → HK/K?)(d) Rather than assuming that K is a normal subgroup, suppose that we only assume that HCN(K); i.e., we assume that H is contained in the normalizer of K. Prove that (a), (b), and (c)are true.

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Answered: (c) Prove that HK/K is isomorphic to…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 23E

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(c) Prove that HK/K is isomorphic to H/(H ^ K). (Hint. What is the kernel of the surjective homomorphism H → HK/K?)