STRUCTURE OF ALGEBRA
(Homomorphism and Factor Group)
Drs. Antonius Cahya Prihandoko, M.App.Sc
Mathematics Education Study Program Faculty of Teacher Training and Education
the University of Jember Indonesia
Outline
1 Homomorphism
Properties of Homomorphism Isomorphism and Cayley’s Theorem
2 Factor Groups
Outline
1 Homomorphism
Properties of Homomorphism Isomorphism and Cayley’s Theorem
2 Factor Groups
Definition of Homomorphism
Definition
A mapφfrom group(G,∗)into group(G′,#)is called homomorphismif
φ(a∗b) =φ(a)#φ(b)
for allaandbinG.
Example 1
Definition of Homomorphism
Definition
A mapφfrom group(G,∗)into group(G′,#)is called homomorphismif
φ(a∗b) =φ(a)#φ(b)
for allaandbinG.
Example 1
Letα:Z → Zndefined byα(m) =r, wherer is the residue
Definition of Homomorphism
Example 2
LetSnis symmetric group ofnletters, and letφ:Sn→ Z2
defined by
φ(ρ) =
0 ifρis even permutation,
1 ifρis odd permutation
Show thatφis a homomorphism!
Example 3
Definition of Homomorphism
Example 2
LetSnis symmetric group ofnletters, and letφ:Sn→ Z2
defined by
φ(ρ) =
0 ifρis even permutation,
1 ifρis odd permutation
Show thatφis a homomorphism!
Example 3
Some Properties
Fundamental Properties
Letφis a homomorphism from groupg into groupG′, then :
1 Ifeis the identity element ofG, thenφ(e)is the identity element ofG′;
2 Ifa∈G, thenφ(a−1) =φ(a)−1;
3 IfH is subgroup ofG, thenφ(H)is subgroup ofG′;
Some Properties
Fundamental Properties
Letφis a homomorphism from groupg into groupG′, then :
1 Ifeis the identity element ofG, thenφ(e)is the identity
element ofG′;
2 Ifa∈G, thenφ(a−1) =φ(a)−1;
3 IfH is subgroup ofG, thenφ(H)is subgroup ofG′;
Some Properties
Fundamental Properties
Letφis a homomorphism from groupg into groupG′, then :
1 Ifeis the identity element ofG, thenφ(e)is the identity
element ofG′;
2 Ifa∈G, thenφ(a−1) =φ(a)−1;
3 IfH is subgroup ofG, thenφ(H)is subgroup ofG′;
Some Properties
Fundamental Properties
Letφis a homomorphism from groupg into groupG′, then :
1 Ifeis the identity element ofG, thenφ(e)is the identity
element ofG′;
2 Ifa∈G, thenφ(a−1) =φ(a)−1;
3 IfH is subgroup ofG, thenφ(H)is subgroup ofG′;
Some Properties
Fundamental Properties
Letφis a homomorphism from groupg into groupG′, then :
1 Ifeis the identity element ofG, thenφ(e)is the identity
element ofG′;
2 Ifa∈G, thenφ(a−1) =φ(a)−1;
3 IfH is subgroup ofG, thenφ(H)is subgroup ofG′;
Kernel
Definition of Kernel
Letφ:G→G′ is a homomorphism, thenkernelofφ, denoted
byKer(φ), is defined as
Ker(φ) =φ−1({
e′}) ={a∈G|φ(a) =e′}
wheree′ is the identity element ofG′.
Kernel as a subgroup
Ifφ:G→G′ is a homomorphism, thenKer(φ)is a subgroup of
Kernel
Definition of Kernel
Letφ:G→G′ is a homomorphism, thenkernelofφ, denoted
byKer(φ), is defined as
Ker(φ) =φ−1({
e′}) ={a∈G|φ(a) =e′}
wheree′ is the identity element ofG′.
Kernel as a subgroup
Ifφ:G→G′ is a homomorphism, thenKer(φ)is a subgroup of
Kernel
Cosets of the Kernel
Letφ:G→G′ is a homomorphism andH =Ker(φ). Leta∈G.
Then
φ−1{φ(a)}={x ∈G|φ(x) =φ(a)}
is the left coset,aH, and also the right coset,Ha.
Corollary
A homomorphismφ:G→G′ is one-to-one function if and only
Kernel
Cosets of the Kernel
Letφ:G→G′ is a homomorphism andH =Ker(φ). Leta∈G.
Then
φ−1{φ(a)}={x ∈G|φ(x) =φ(a)}
is the left coset,aH, and also the right coset,Ha.
Corollary
A homomorphismφ:G→G′ is one-to-one function if and only
Some Terminologies
On homomorphism
Letφ:G→G′ is a homomorphism. Ifφis one-to-one then it is
called monomorphism. Ifφis onto then it is called
epimorphism. Ifφis both one-to-one and onto, it is called isomorphism
On Subgroup
Some Terminologies
On homomorphism
Letφ:G→G′ is a homomorphism. Ifφis one-to-one then it is
called monomorphism. Ifφis onto then it is called
epimorphism. Ifφis both one-to-one and onto, it is called isomorphism
On Subgroup
A subgroupHof groupG)can be callednormalwhen
Isomorphism
Definition
Anisomorphismφ:G→G′ is one-to-one homomorphism
fromGontoG′. The notation for two groups that are isomorph
isG≃G′.
Theorem
Letζ is a set of groups. ForGandG′ both are inζ, we say
G≃G′, if they are isomorph. Then≃is an equivalence
Isomorphism
Definition
Anisomorphismφ:G→G′ is one-to-one homomorphism
fromGontoG′. The notation for two groups that are isomorph
isG≃G′.
Theorem
Letζ is a set of groups. ForGandG′ both are inζ, we say
G≃G′, if they are isomorph. Then≃is an equivalence
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
All finite cyclic group,G, are isomorph to the group of integers,
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG
intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
All finite cyclic group,G, are isomorph to the group of integers,
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG
intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
All finite cyclic group,G, are isomorph to the group of integers,
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG
intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG
intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
Isomorphism
How to show two groups are isomorph?
1 Define a functionφas a candidate of isomorphism fromG
intoG′.
2 Show thatφis one-to-one function.
3 Show thatφis onto.
4 Show thatφ(xy) =φ(x)φ(y),∀x,y ∈G.
Theorem
All finite cyclic group,G, are isomorph to the group of integers,
Structural Properties
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; Infinity;
Structural Properties
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; Infinity;
Structural Properties
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;
Infinity;
Structural Properties
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; Infinity;
Structural Properties
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; Infinity;
Order of group and order of element;
Structural Properties
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; Infinity;
Order of group and order of element; Number of elements of certain order.
Structural Properties
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; Infinity;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers; Binary operation on a group;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers; Binary operation on a group;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers;
Binary operation on a group;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers; Binary operation on a group;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers; Binary operation on a group;
All group elements are permutations;
Non Structural Properties
On the other side
An isomorphism does not keep non structural properties. Such properties are:
Group containing 5;
All group elements are numbers; Binary operation on a group;
Cayley’s Theorem
Cayley’s Theorem
Every group is isomorph to a group of permutations.
How to prove that theorem?
1 Start with a groupG, determine a permutations setG′
2 Prove thatG′ forms a group under permutation
multiplication.
Cayley’s Theorem
Cayley’s Theorem
Every group is isomorph to a group of permutations.
How to prove that theorem?
1 Start with a groupG, determine a permutations setG′
2 Prove thatG′ forms a group under permutation multiplication.
3 Define a functionφ:G→G′and show thatφis an
Cayley’s Theorem
Cayley’s Theorem
Every group is isomorph to a group of permutations.
How to prove that theorem?
1 Start with a groupG, determine a permutations setG′
2 Prove thatG′ forms a group under permutation multiplication.
Cayley’s Theorem
Cayley’s Theorem
Every group is isomorph to a group of permutations.
How to prove that theorem?
1 Start with a groupG, determine a permutations setG′
2 Prove thatG′ forms a group under permutation
multiplication.
Cayley’s Theorem
Cayley’s Theorem
Every group is isomorph to a group of permutations.
How to prove that theorem?
1 Start with a groupG, determine a permutations setG′
2 Prove thatG′ forms a group under permutation
multiplication.
3 Define a functionφ:G→G′and show thatφis an
Factor Groups
Coming from Homomorphism
Letφ:G→G′ is a homomorphism withKer(φ) =H. Then
R/H ={a∗H|a∈R}is a group under binary operation:
(aH)(bH) = (ab)H
Factor Groups
Coming from Normal Subgroup
LetHis a subgroup of groupG. Multiplication on cosets ofHis defined as(aH)(bH) = (ab)Hiswell-definedif and only
aH=Ha,∀a∈G
Coming from Normal Subgroup
As consequence.LetHis a normal subgroup normal on group G. ThenG/H ={aH|a∈G}is a group under binary operation
Factor Groups
Coming from Normal Subgroup
LetHis a subgroup of groupG. Multiplication on cosets ofHis defined as(aH)(bH) = (ab)Hiswell-definedif and only
aH=Ha,∀a∈G
Coming from Normal Subgroup
As consequence.LetHis a normal subgroup normal on group
G. ThenG/H ={aH|a∈G}is a group under binary operation
Characterize of a Normal Subgroup
Equivalence Characteristics
For a normal subgroupH in groupG.
1 ghg−1∈H,∀g∈Gdanh∈H.
2 gHg−1=H,∀g ∈G.
3 gH =Hg,∀g ∈G.
Automorphism
An isomorphismφ:G→Gis calledautomorphisminG. AutomorphismIg :G→GwhereIg(x) =gxg−1is calledinner
Characterize of a Normal Subgroup
Equivalence Characteristics
For a normal subgroupH in groupG.
1 ghg−1∈H,∀g∈Gdanh∈H.
2 gHg−1=H,∀g ∈G.
3 gH =Hg,∀g ∈G.
Automorphism
An isomorphismφ:G→Gis calledautomorphisminG. AutomorphismIg :G→GwhereIg(x) =gxg−1is calledinner
Characterize of a Normal Subgroup
Equivalence Characteristics
For a normal subgroupH in groupG.
1 ghg−1∈H,∀g∈Gdanh∈H.
2 gHg−1=H,∀g ∈G.
3 gH =Hg,∀g ∈G.
Automorphism
Characterize of a Normal Subgroup
Equivalence Characteristics
For a normal subgroupH in groupG.
1 ghg−1∈H,∀g∈Gdanh∈H.
2 gHg−1=H,∀g ∈G.
3 gH =Hg,∀g ∈G.
Automorphism
Characterize of a Normal Subgroup
Equivalence Characteristics
For a normal subgroupH in groupG.
1 ghg−1∈H,∀g∈Gdanh∈H.
2 gHg−1=H,∀g ∈G.
3 gH =Hg,∀g ∈G.
Automorphism
An isomorphismφ:G→Gis calledautomorphisminG.
Fundamental Homomorphism Theorem
Prelimenary
LetHis a normal subgroup of groupG. Thenφ:G→G/H, defined byφ(a) =aH, is a homomorphism with the kernel
Ker(φ) =H.
Fundamental Homomorphism Theorem
Letφ:G→G′ be a homomorphism withKer(φ) =H. Then
Fundamental Homomorphism Theorem
Prelimenary
LetHis a normal subgroup of groupG. Thenφ:G→G/H, defined byφ(a) =aH, is a homomorphism with the kernel
Ker(φ) =H.
Fundamental Homomorphism Theorem
Letφ:G→G′ be a homomorphism withKer(φ) =H. Then φ(G)is a group, and mappingµ:G/H→φ(G), defined by
µ(aH) =φ(a), is a isomorphism. Ifγ :G→G/Hbe a homomorphism defined byγ(a) =aH, then∀a∈G,
