Let G be a group and N a normal subgroup of G. Consider the map : G→ G/N defined by 4(g)=gN. 1. Prove that is a homomorphism. 2. Show that the kernel of p is exactly 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 3D3, we really mean that the structure of 73 appears in D3. This can be formalized by a map : Z3D3, defined by : nr. AGL1(Z5) Dic 10 C10 C C C C CA Z3 (r) (r) (12) C2 C2 C2 C2 C2 C₁ 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 : G H 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)= {(9) | 9€ G}.
Let G be a group and N a normal subgroup of G. Consider the map : G→ G/N defined by 4(g)=gN. 1. Prove that is a homomorphism. 2. Show that the kernel of p is exactly 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 3D3, we really mean that the structure of 73 appears in D3. This can be formalized by a map : Z3D3, defined by : nr. AGL1(Z5) Dic 10 C10 C C C C CA Z3 (r) (r) (12) C2 C2 C2 C2 C2 C₁ 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 : G H 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)= {(9) | 9€ G}.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.2: Cayley’s Theorem
Problem 12E: Find the right regular representation of G as defined Exercise 11 for each of the following groups....
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Transcribed Image Text:Let G be a group and N a normal subgroup of G. Consider the map : G→ G/N defined by
4(g)=gN.
1. Prove that is a homomorphism.
2. Show that the kernel of p is exactly 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 3D3, we really mean that the structure of 73 appears in D3.
This can be formalized by a map : Z3D3, defined by : nr.
AGL1(Z5)
Dic 10
C10
C C C C CA
Z3
(r)
(r) (12)
C2 C2
C2 C2 C2
C₁
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 : G H 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)= {(9) | 9€ G}.
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