STRUCTURE OF ALGEBRA (Homomorphism and Factor Group)
STRUCTURE OF ALGEBRA
(Homomorphism and Factor Group)
Drs. Antonius Cahya Prihandoko, M.App.Sc
Definition ′
A map φ from group (G, ∗) into group (G , #) is called
homomorphism if
φ(a ∗ b) = φ(a)#φ(b) Definition ′
A map φ from group (G, ∗) into group (G , #) is called
homomorphism if
φ(a ∗ b) = φ(a)#φ(b) Example 2
Let S is symmetric group of n letters, and let φ : S → Z
n n
2
defined by if ρ is even permutation, φ(ρ) = 1 if
ρ is odd permutation Example 2
Let S is symmetric group of n letters, and let φ : S → Z
n n
2
defined by if ρ is even permutation, φ(ρ) = 1 if
ρ is odd permutation Fundamental Properties ′
Let , then : 1 φ is a homomorphism from group g into group G
If e is the identity element of G, then φ(e) is the identity ′ element of G ; Fundamental Properties ′
Let , then : 1 φ is a homomorphism from group g into group G
If e is the identity element of G, then φ(e) is the identity
′
element of G ; Fundamental Properties ′
Let , then : 1 φ is a homomorphism from group g into group G
If e is the identity element of G, then φ(e) is the identity
′
element of G ; Fundamental Properties ′
Let , then : 1 φ is a homomorphism from group g into group G
If e is the identity element of G, then φ(e) is the identity
′
element of G ; Fundamental Properties ′
Let , then : 1 φ is a homomorphism from group g into group G
If e is the identity element of G, then φ(e) is the identity
′
element of G ; Definition of Kernel ′
Let is a homomorphism, then kernel of φ : G → G φ, denoted by Ker
(φ), is defined as
−1 ′ ′ Ker
(φ) = φ ({e }) = {a ∈ G|φ(a) = e } Definition of Kernel ′
Let is a homomorphism, then kernel of φ : G → G φ, denoted by Ker
(φ), is defined as
−1 ′ ′ Ker
(φ) = φ ({e }) = {a ∈ G|φ(a) = e } Cosets of the Kernel ′ Let φ : G → G is a homomorphism and H = Ker (φ). Let a ∈ G.
Then
−1
φ {φ(a)} = {x ∈ G|φ(x) = φ(a)} is the left coset, aH, and also the right coset, Ha. Cosets of the Kernel ′ Let φ : G → G is a homomorphism and H = Ker (φ). Let a ∈ G.
Then
−1
φ {φ(a)} = {x ∈ G|φ(x) = φ(a)} is the left coset, aH, and also the right coset, Ha. On homomorphism ′
Let is a homomorphism. If φ : G → G φ is one-to-one then it is called monomorphism. If
φ is onto then it is called epimorphism. If φ is both one-to-one and onto, it is called isomorphism On homomorphism ′
Let is a homomorphism. If φ : G → G φ is one-to-one then it is called monomorphism. If
φ is onto then it is called epimorphism. If φ is both one-to-one and onto, it is called isomorphism Definition
′
An isomorphism is one-to-one homomorphism φ : G → G
′
from G onto G . The notation for two groups that are isomorph
′ is G .
≃ G Definition
′
An isomorphism is one-to-one homomorphism φ : G → G
′
from G onto G . The notation for two groups that are isomorph
′ is G .
≃ G How to show two groups are isomorph? 1 Define a function φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are isomorph? 1 Define a function
φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are isomorph? 1 Define a function
φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are isomorph? 1 Define a function
φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are isomorph? 1 Define a function
φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are isomorph? 1 Define a function
φ as a candidate of isomorphism from G
′ 2 into G .
Show that 3 φ is one-to-one function.
Show that φ is onto. How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below:
Cyclic; Commute; How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic;
Commute; How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic; Commute;
How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic; Commute;
How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic; Commute;
How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic; Commute;
How to show two groups are not isomorph
We have to know that an isomorphism keeps structural properties of a group. Such properties can be shown below: Cyclic; Commute;
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5; All group elements are numbers; On the other side
An isomorphism does not keep non structural properties. Such properties are: Group containing 5;
All group elements are numbers; On the other side
An isomorphism does not keep non structural properties. Such properties are: Group containing 5; All group elements are numbers;
On the other side
An isomorphism does not keep non structural properties. Such properties are: Group containing 5; All group elements are numbers;
On the other side
An isomorphism does not keep non structural properties. Such properties are: Group containing 5; All group elements are numbers;
On the other side
An isomorphism does not keep non structural properties. Such properties are: Group containing 5; All group elements are numbers;
Cayley’s Theorem Every group is isomorph to a group of permutations.
How to prove that theorem? 1 ′
Start with a group G, determine a permutations set G Cayley’s Theorem Every group is isomorph to a group of permutations. How to prove that theorem? 1
′ Start with a group G, determine a permutations set G Cayley’s Theorem Every group is isomorph to a group of permutations. How to prove that theorem? 1
′
Start with a group G, determine a permutations set G Cayley’s Theorem Every group is isomorph to a group of permutations. How to prove that theorem? 1
′
Start with a group G, determine a permutations set G Cayley’s Theorem Every group is isomorph to a group of permutations. How to prove that theorem? 1
′
Start with a group G, determine a permutations set G Coming from Homomorphism ′
Let is a homomorphism with Ker φ : G → G (φ) = H. Then
R
/H = {a ∗ H|a ∈ R} is a group under binary operation: Coming from Normal Subgroup
Let H is a subgroup of group G. Multiplication on cosets of H is defined as (aH)(bH) = (ab)H is well-defined if and only
aH
= Ha, ∀a ∈ G Coming from Normal Subgroup
Let H is a subgroup of group G. Multiplication on cosets of H is defined as (aH)(bH) = (ab)H is well-defined if and only
aH
= Ha, ∀a ∈ G Equivalence Characteristics For a normal subgroup H in group G. 1 −1 ghg ∈ H, ∀g ∈ G dan h ∈ H. 2 −1 gHg 3 = H, ∀g ∈ G. gH = Hg, ∀g ∈ G.
Equivalence Characteristics For a normal subgroup H in group G. 1 −1 ghg ∈ H, ∀g ∈ G dan h ∈ H. 2 −1 gHg 3 = H, ∀g ∈ G. gH = Hg, ∀g ∈ G.
Equivalence Characteristics For a normal subgroup H in group G. 1 −1 ghg ∈ H, ∀g ∈ G dan h ∈ H. 2 −1 gHg 3 = H, ∀g ∈ G. gH = Hg, ∀g ∈ G.
Equivalence Characteristics For a normal subgroup H in group G. 1 −1 ghg ∈ H, ∀g ∈ G dan h ∈ H. 2 −1 gHg 3 = H, ∀g ∈ G. gH = Hg, ∀g ∈ G.
Equivalence Characteristics For a normal subgroup H in group G. 1 −1 ghg ∈ H, ∀g ∈ G dan h ∈ H. 2 −1 gHg 3 = H, ∀g ∈ G. gH = Hg, ∀g ∈ G.
Prelimenary
Let H is a normal subgroup of group G. Then φ : G → G/H, defined by φ(a) = aH, is a homomorphism with the kernel
Ker (φ) = H.
Fundamental Homomorphism Theorem
Prelimenary
Let H is a normal subgroup of group G. Then φ : G → G/H, defined by φ(a) = aH, is a homomorphism with the kernel
Ker (φ) = H.
Fundamental Homomorphism Theorem