Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G + H is a homomorphism, then Im(4)G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group, H a normal subgroup of G, and K a subgroup of G. Show that the map :G/H→G/K induced by the natural projection maps is a homomorphism, and further prove that it is injective if and only if H = K. Hint: Apply the Fundamental Homomorphism Theorem for the maps G→G/H and G→G/K, and analyze the injectivity condition. Visualizing the FHT via Cayley graphs G (Ker()G) any homomorphism quotient process G/Ker group of cosets "K Im(6) H Q8 iN remaining isomorphism ("relabeling") "quotient map" N $ = LOT QB/N N iN kN VA "relabeling map"

Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G + H is a homomorphism, then Im(4)G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group, H a normal subgroup of G, and K a subgroup of G. Show that the map :G/H→G/K induced by the natural projection maps is a homomorphism, and further prove that it is injective if and only if H = K. Hint: Apply the Fundamental Homomorphism Theorem for the maps G→G/H and G→G/K, and analyze the injectivity condition. Visualizing the FHT via Cayley graphs G (Ker()G) any homomorphism quotient process G/Ker group of cosets "K Im(6) H Q8 iN remaining isomorphism ("relabeling") "quotient map" N $ = LOT QB/N N iN kN VA "relabeling map"

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 4E: 4. Prove that the special linear group is a normal subgroup of the general linear group .

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