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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?)

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 normal subgroup 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 surjective homomorphism H → HK/K?) (d) Rather than assuming that K is a normal subgroup, suppose that we only assume that HC N(K); i.e., we assume that H is contained in the normalizer of K. Prove that (a), (b), and (c) are true.
Transcribed Image Text:6.6. Let G be a group, let H C G and K C G be subgroups, and assume that K is a normal subgroup 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 surjective homomorphism H → HK/K?) (d) Rather than assuming that K is a normal subgroup, suppose that we only assume that HC N(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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