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Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G H 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 group, X a set, and pp: G→ Sym(X) a homomorphism, where Sym(X) is the symmetric group on X. Prove that if ker() = {e}, then the map p is injective, and hence G is isomorphic to a subgroup of Sym(X). Hint: Use the Fundamental Homomorphism Theoren and the fact that injectivity of implies that G is isomorphic to its image, which is a subgroup of Sym(X). Proof approach: Visualizing the FHT via Cayley graphs G In(d) SH (Ker()G) any homomorphism quotient process G/Kerle group of cosets remaining isomorphism ("relabeling") QB iN IN kN "quotient map" N = LOT QB/N jN 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()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, X a set, and pp: G→ Sym(X) a homomorphism, where Sym(X) is the symmetric group on X. Prove that if ker() = {e}, then the map p is injective, and hence G is isomorphic to a subgroup of Sym(X). Hint: Use the Fundamental Homomorphism Theoren and the fact that injectivity of implies that G is isomorphic to its image, which is a subgroup of Sym(X). Proof approach: Visualizing the FHT via Cayley graphs G In(d) SH (Ker()G) any homomorphism quotient process G/Kerle group of cosets remaining isomorphism ("relabeling") QB iN IN kN "quotient map" N = LOT QB/N jN iN KN VA "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 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...
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Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G H 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 group, X a set, and pp: G→ Sym(X) a homomorphism, where Sym(X) is the symmetric group on X. Prove that if ker() = {e}, then the map p is injective, and hence G is isomorphic to a subgroup of Sym(X). Hint: Use the Fundamental Homomorphism Theoren and the fact that injectivity of implies that G is isomorphic to its image, which is a subgroup of Sym(X). Proof approach: Visualizing the FHT via Cayley graphs G In(d) SH (Ker()G) any homomorphism quotient process G/Kerle group of cosets remaining isomorphism ("relabeling") QB iN IN kN "quotient map" N = LOT QB/N jN iN KN VA "relabeling map"
Transcribed Image Text:Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G H 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 group, X a set, and pp: G→ Sym(X) a homomorphism, where Sym(X) is the symmetric group on X. Prove that if ker() = {e}, then the map p is injective, and hence G is isomorphic to a subgroup of Sym(X). Hint: Use the Fundamental Homomorphism Theoren and the fact that injectivity of implies that G is isomorphic to its image, which is a subgroup of Sym(X). Proof approach: Visualizing the FHT via Cayley graphs G In(d) SH (Ker()G) any homomorphism quotient process G/Kerle group of cosets remaining isomorphism ("relabeling") QB iN IN kN "quotient map" N = LOT QB/N jN iN KN VA "relabeling map"
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