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Let G be a group and H a subgroup of G. 1. Prove that the inclusion map: HG, where (h) = h for all h€ H, is an embedding if and only if H is a subgroup of G and is injective. 2. Let N be a normal subgroup of G. Prove that the quotient map q: G→ G/N is surjective and that ker(q) = N. The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say Z3 D3. we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3D3. defined by : nr. AGL1(Z5) C10 Dic 10 GGG C C C C CA Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. Z3 (r) (f) (m) (P) 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($) = $(G) = {$(g) | g € G}.

Let G be a group and H a subgroup of G. 1. Prove that the inclusion map: HG, where (h) = h for all h€ H, is an embedding if and only if H is a subgroup of G and is injective. 2. Let N be a normal subgroup of G. Prove that the quotient map q: G→ G/N is surjective and that ker(q) = N. The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say Z3 D3. we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3D3. defined by : nr. AGL1(Z5) C10 Dic 10 GGG C C C C CA Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. Z3 (r) (f) (m) (P) 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($) = $(G) = {$(g) | g € 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
Section4.6: Quotient Groups
Problem 23E
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Let G be a group and H a subgroup of G. 1. Prove that the inclusion map: HG, where (h) = h for all h€ H, is an embedding if and only if H is a subgroup of G and is injective. 2. Let N be a normal subgroup of G. Prove that the quotient map q: G→ G/N is surjective and that ker(q) = N. The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say Z3 D3. we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3D3. defined by : nr. AGL1(Z5) C10 Dic 10 GGG C C C C CA Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. Z3 (r) (f) (m) (P) 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($) = $(G) = {$(g) | g € G}.
Transcribed Image Text:Let G be a group and H a subgroup of G. 1. Prove that the inclusion map: HG, where (h) = h for all h€ H, is an embedding if and only if H is a subgroup of G and is injective. 2. Let N be a normal subgroup of G. Prove that the quotient map q: G→ G/N is surjective and that ker(q) = N. The difference between embeddings and quotient maps can be seen in the subgroup lattice: When we say Z3 D3. we really mean that the structure of 3 appears in D3. This can be formalized by a map : Z3D3. defined by : nr. AGL1(Z5) C10 Dic 10 GGG C C C C CA Cz Cz C2 C2 C2 In one of these groups, D5 is subgroup. In the other, it arises as a quotient. Z3 (r) (f) (m) (P) 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($) = $(G) = {$(g) | g € G}.
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