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 p. Let G be a group and H a normal subgroup of G. Suppose : G/H→ K is a homomorphism. Prove that there exists a homomorphism : G→ K such that (gH) = (gH) for all g€ G, and that is uniquely determined by p. Hint: Use the universal property of quotient groups and the First Isomorphism Theorem to define and prove the existence of . Visualizing the FHT via Cayley graphs G Im(6) H (Ker()G) any homomorphism quotient process G/Ker() group of cosets QB IN kN remaining isomorphism "quotient map" ("relabeling") N jN $ = LOπ Qo/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 p. Let G be a group and H a normal subgroup of G. Suppose : G/H→ K is a homomorphism. Prove that there exists a homomorphism : G→ K such that (gH) = (gH) for all g€ G, and that is uniquely determined by p. Hint: Use the universal property of quotient groups and the First Isomorphism Theorem to define and prove the existence of . Visualizing the FHT via Cayley graphs G Im(6) H (Ker()G) any homomorphism quotient process G/Ker() group of cosets QB IN kN remaining isomorphism "quotient map" ("relabeling") N jN $ = LOπ Qo/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

Section4.6: Quotient Groups

Problem 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...

See similar textbooks