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, and let N and M be normal subgroups of G. Prove that if N is isomorphic to M as groups, then G/N is isomorphic to G/M. Hint: Use the Fundamental Homomorphism Theorem to define an isomorphism between the two quotient groups. Visualizing the FHT via Cayley graphs G Im(6) H (Ker(0) 4G) any homomorphism quotient process G/Karld group of cosets remaining isomorphism ("relabeling") iN QU "quotient map" π kN jN $ = LOTT Qo/N iN kN) h 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, and let N and M be normal subgroups of G. Prove that if N is isomorphic to M as groups, then G/N is isomorphic to G/M. Hint: Use the Fundamental Homomorphism Theorem to define an isomorphism between the two quotient groups. Visualizing the FHT via Cayley graphs G Im(6) H (Ker(0) 4G) any homomorphism quotient process G/Karld group of cosets remaining isomorphism ("relabeling") iN QU "quotient map" π kN jN $ = LOTT Qo/N iN kN) h 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