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Let G be a group, and let N be a normal subgroup of G. 1. Prove that the map : GG/N, given by (g)=gN, is a group homomorphism. 2. Show that is surjective and that ker() = N. 3. Using the first isomorphism theorem, conclude that G/NG/ker(). The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say 3D3, we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3 D3, defined by : nr. AGL1(Z6) GG C10 Dic 10 Z3 (r) in imien (f) (rf) (r²f) (0) <1> Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. This, and much more, will be consequences of the celebrated isomorphism theorems. In general, a homomomorphism is a function : GH with some extra properties. We will use standard function terminology: ■the group G is the domain ■the group H is the codomain the image is what is often called the range: Im(d)(G){$(g) | 9€ G}.

Let G be a group, and let N be a normal subgroup of G. 1. Prove that the map : GG/N, given by (g)=gN, is a group homomorphism. 2. Show that is surjective and that ker() = N. 3. Using the first isomorphism theorem, conclude that G/NG/ker(). The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say 3D3, we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3 D3, defined by : nr. AGL1(Z6) GG C10 Dic 10 Z3 (r) in imien (f) (rf) (r²f) (0) <1> Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. This, and much more, will be consequences of the celebrated isomorphism theorems. In general, a homomomorphism is a function : GH with some extra properties. We will use standard function terminology: ■the group G is the domain ■the group H is the codomain the image is what is often called the range: Im(d)(G){$(g) | 9€ G}.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 5E: 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that...
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Let G be a group, and let N be a normal subgroup of G. 1. Prove that the map : GG/N, given by (g)=gN, is a group homomorphism. 2. Show that is surjective and that ker() = N. 3. Using the first isomorphism theorem, conclude that G/NG/ker(). The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say 3D3, we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3 D3, defined by : nr. AGL1(Z6) GG C10 Dic 10 Z3 (r) in imien (f) (rf) (r²f) (0) <1> Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. This, and much more, will be consequences of the celebrated isomorphism theorems. In general, a homomomorphism is a function : GH with some extra properties. We will use standard function terminology: ■the group G is the domain ■the group H is the codomain the image is what is often called the range: Im(d)(G){$(g) | 9€ G}.
Transcribed Image Text:Let G be a group, and let N be a normal subgroup of G. 1. Prove that the map : GG/N, given by (g)=gN, is a group homomorphism. 2. Show that is surjective and that ker() = N. 3. Using the first isomorphism theorem, conclude that G/NG/ker(). The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say 3D3, we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3 D3, defined by : nr. AGL1(Z6) GG C10 Dic 10 Z3 (r) in imien (f) (rf) (r²f) (0) <1> Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. This, and much more, will be consequences of the celebrated isomorphism theorems. In general, a homomomorphism is a function : GH with some extra properties. We will use standard function terminology: ■the group G is the domain ■the group H is the codomain the image is what is often called the range: Im(d)(G){$(g) | 9€ G}.
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