Let G be a group and N a normal subgroup of G. 1. Prove that the map p: G G/N, defined by (g)=gN, is a surjective homomorphism. 2. Show that the kernel of p is N. 3. Use the first isomorphism theorem to conclude that G/NG/ker(). 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 Z3 appears in D3. This can be formalized by a map : Z3D3, defined by : nr. AGL1(Z5) C10 Dic 10 C G C G Z3 (f) (rf) (°f) (0) (1) C2 C2 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(4) (G) (9) 9 G}.
Let G be a group and N a normal subgroup of G. 1. Prove that the map p: G G/N, defined by (g)=gN, is a surjective homomorphism. 2. Show that the kernel of p is N. 3. Use the first isomorphism theorem to conclude that G/NG/ker(). 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 Z3 appears in D3. This can be formalized by a map : Z3D3, defined by : nr. AGL1(Z5) C10 Dic 10 C G C G Z3 (f) (rf) (°f) (0) (1) C2 C2 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(4) (G) (9) 9 G}.
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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Transcribed Image Text:Let G be a group and N a normal subgroup of G.
1. Prove that the map p: G
G/N, defined by (g)=gN, is a surjective homomorphism.
2. Show that the kernel of p is N.
3. Use the first isomorphism theorem to conclude that G/NG/ker().
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 Z3 appears in D3.
This can be formalized by a map : Z3D3, defined by : nr.
AGL1(Z5)
C10
Dic 10
C G C G
Z3
(f) (rf) (°f)
(0)
(1)
C2 C2
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(4) (G) (9) 9 G}.
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