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 finite group and N a normal subgroup of G. Define the map : G G/N as the natural projection homomorphism. Prove that induces a faithful group action of G on the set G/N, and show that the kernel of this action is exactly N. Hint: Use the First Isomorphism Theorem and properties of normal subgroups to explore the kernel of the action. Visualizing the FHT via Cayley graphs G Im(0) SH (Ker(6) G) any homomorphism quotient process G/Ker(d) group of cosets QB kN remaining isomorphism "quotient map" ("relabeling") N jN $ = LOTT Qo/N VA "relabeling map" iN kN

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 finite group and N a normal subgroup of G. Define the map : G G/N as the natural projection homomorphism. Prove that induces a faithful group action of G on the set G/N, and show that the kernel of this action is exactly N. Hint: Use the First Isomorphism Theorem and properties of normal subgroups to explore the kernel of the action. Visualizing the FHT via Cayley graphs G Im(0) SH (Ker(6) G) any homomorphism quotient process G/Ker(d) group of cosets QB kN remaining isomorphism "quotient map" ("relabeling") N jN $ = LOTT Qo/N VA "relabeling map" iN kN

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 11E: Find all homomorphic images of the quaternion group.

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