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Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : GH is a homomorphism, then Im($) ≈ G/Ker($). 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. Prove that the order of the quotient group G/N is given by the index of N in G, i.e., |G| |G|N|= |N| Visualizing the FHT via Cayley graphs G N Im() SH Ker(G) any homomorphism jN quotient process G/Ker() group of cosets remaining isomorphism (relabeling QB "quotient map" ㅠ iN KN N jN Φ VA $ = =40π QB/N ¡N KN "relabeling map"

Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : GH is a homomorphism, then Im($) ≈ G/Ker($). 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. Prove that the order of the quotient group G/N is given by the index of N in G, i.e., |G| |G|N|= |N| Visualizing the FHT via Cayley graphs G N Im() SH Ker(G) any homomorphism jN quotient process G/Ker() group of cosets remaining isomorphism (relabeling QB "quotient map" ㅠ iN KN N jN Φ VA $ = =40π QB/N ¡N KN "relabeling map"

Elements Of Modern Algebra
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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Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : GH is a homomorphism, then Im($) ≈ G/Ker($). 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. Prove that the order of the quotient group G/N is given by the index of N in G, i.e., |G| |G|N|= |N| Visualizing the FHT via Cayley graphs G N Im() SH Ker(G) any homomorphism jN quotient process G/Ker() group of cosets remaining isomorphism (relabeling QB "quotient map" ㅠ iN KN N jN Φ VA $ = =40π QB/N ¡N KN "relabeling map"
Transcribed Image Text:Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : GH is a homomorphism, then Im($) ≈ G/Ker($). 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. Prove that the order of the quotient group G/N is given by the index of N in G, i.e., |G| |G|N|= |N| Visualizing the FHT via Cayley graphs G N Im() SH Ker(G) any homomorphism jN quotient process G/Ker() group of cosets remaining isomorphism (relabeling QB "quotient map" ㅠ iN KN N jN Φ VA $ = =40π QB/N ¡N KN "relabeling map"
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