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. Suppose G and H are groups, and that ф : G-> H is a homomorphism. Let K denote Ker(O). Prove that if αι A's a2K for solne α 1.126 G, then φ(al)-0(a2). 1. 2. The fact you proved in part (a) allows us to define a new mapping e: G/K-+ H by (aK)-d(a) for all a K G/K. Prove that ψ is a homomorphism fron G/K to H Definition: Suppose that G and H are groups and that φ : G → H is a homomorphism. The kernel of φ, denoted by Ker(о), is defined to be the set {g Glo(g)-ell), where e" is the identity of H Definitions: . Suppose that (G,*) and (1,0) are groups. We say that a function φ : G → H is a hornomorphism if for all

. Suppose G and H are groups, and that ф : G-> H is a homomorphism. Let K denote Ker(O). Prove that if αι A's a2K for solne α 1.126 G, then φ(al)-0(a2). 1. 2. The fact you proved in part (a) allows us to define a new mapping e: G/K-+ H by (aK)-d(a) for all a K G/K. Prove that ψ is a homomorphism fron G/K to H Definition: Suppose that G and H are groups and that φ : G → H is a homomorphism. The kernel of φ, denoted by Ker(о), is defined to be the set {g Glo(g)-ell), where e" is the identity of H Definitions: . Suppose that (G,*) and (1,0) are groups. We say that a function φ : G → H is a hornomorphism if for all

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 31E: A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements...
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. Suppose G and H are groups, and that ф : G-> H is a homomorphism. Let K denote Ker(O). Prove that if αι A's a2K for solne α 1.126 G, then φ(al)-0(a2). 1. 2. The fact you proved in part (a) allows us to define a new mapping e: G/K-+ H by (aK)-d(a) for all a K G/K. Prove that ψ is a homomorphism fron G/K to H Definition: Suppose that G and H are groups and that φ : G → H is a homomorphism. The kernel of φ, denoted by Ker(о), is defined to be the set {g Glo(g)-ell), where e" is the identity of H Definitions: . Suppose that (G,*) and (1,0) are groups. We say that a function φ : G → H is a hornomorphism if for all
Transcribed Image Text:. Suppose G and H are groups, and that ф : G-> H is a homomorphism. Let K denote Ker(O). Prove that if αι A's a2K for solne α 1.126 G, then φ(al)-0(a2). 1. 2. The fact you proved in part (a) allows us to define a new mapping e: G/K-+ H by (aK)-d(a) for all a K G/K. Prove that ψ is a homomorphism fron G/K to H Definition: Suppose that G and H are groups and that φ : G → H is a homomorphism. The kernel of φ, denoted by Ker(о), is defined to be the set {g Glo(g)-ell), where e" is the identity of H Definitions: . Suppose that (G,*) and (1,0) are groups. We say that a function φ : G → H is a hornomorphism if for all
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