Isomorphism
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About This Presentation
This is a handout about Homomorphism, Isomorphism, Kernel and Cayley's Theorem.
This is a group effort.
#LoveforAbstractAlgebra
Slide Content
Page 1 of 3
ISOMORPHISM
GROUP ISOMORPHISM
Let G and G’ be groups with operations ∗and ∗′. An isomorphism ?????? from a group G to a group G’ is a
one – to – one mapping (or function) from G onto G’ that preserves the group operation. That is,
?????? �∗� =?????? � ∗′??????(�) for all a, b in G.
If there is an isomorphism from G onto G’, we say that G and G’ are isomorphic and write G≅ G’.
Steps involved in proving that a group G is isomorphic to group G’:
1. “Mapping.” Define a candidate for isomorphism; that is, define a function ?????? from G to G’.
2. “1 – 1.” Prove that ?????? is one – to – one: that is, assume ?????? � =??????(�) and prove that �=�.
3. “Onto.” Prove that ?????? is onto; that is, from any element g’ in G’, find an element g in G such
that ?????? ?????? =??????′.
4. “O.P.” Prove that ?????? is operation – preserving; that is, show that ?????? �∗� =?????? � ∗′??????(�) for
all a andb in G.
Note: We can relate Steps 2 and 4 to Kernel and Homomorphism, respectively.
HOMOMORPHISM
Let G and G’ be groups with operations ∗and ∗′. A homomorphism of G into G’ is a mapping ?????? of G
intoG’ such that for every a and b in G,
?????? �∗� =?????? � ∗
′
?????? � .
Examples: Determine whether the given map is a homomorphism.
1. Let ??????:�
∗
→�
∗
under addition given by ?????? � =�
2
.
Answer: ?????? �+� = �+�
2
=�
2
+2��+�
2
≠�
2
+�
2
=?????? � +??????(�). Thus ?????? is NOT a
homomorphism.
2. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =�
2
.
Answer: ?????? �� = ��
2
=�
2
�
2
=�
2
�
2
=?????? � ∙??????(�). Thus ?????? is a homomorphism.
3. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =−�.
Answer: ?????? �� =− �� =−��≠ −� (−�)=?????? � ∙??????(�). Thus ?????? is NOT
ahomomorphism.
Types of Homomorphisms:
1. Epimorphism – surjective homomorphism
2. Monomorphism – injective homomorphism
3. Isomorphism – bijective homomorphism
4. Automorphism – isomorphism, domain and codomain are the same group
5. Endomorphism – homomorphism, domain and codomain are the same group
KERNEL
Let ??????:??????→??????′ be a homomorphism of groups. The subgroup ??????
−1
??????
′
= � ?????? ?????? ?????? � =??????′ is the
kernel of ??????, denoted by Ker(??????).
Examples: Find the kernel of the following homomorphism:
1. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =�
2
.
Answer: The identity element of the set of images is 1 under the operation of multiplication.
If �
2
=1 then �=−1 or �=1. Thus, Ker(??????)={−1,1}.
ISOMORPHISM
GROUP ISOMORPHISM
Let G and G’ be groups with operations ∗and ∗′. An isomorphism ?????? from a group G to a group G’ is a
one – to – one mapping (or function) from G onto G’ that preserves the group operation. That is,
?????? �∗� =?????? � ∗′??????(�) for all a, b in G.
If there is an isomorphism from G onto G’, we say that G and G’ are isomorphic and write G≅ G’.
Steps involved in proving that a group G is isomorphic to group G’:
1. “Mapping.” Define a candidate for isomorphism; that is, define a function ?????? from G to G’.
2. “1 – 1.” Prove that ?????? is one – to – one: that is, assume ?????? � =??????(�) and prove that �=�.
3. “Onto.” Prove that ?????? is onto; that is, from any element g’ in G’, find an element g in G such
that ?????? ?????? =??????′.
4. “O.P.” Prove that ?????? is operation – preserving; that is, show that ?????? �∗� =?????? � ∗′??????(�) for
all a andb in G.
Note: We can relate Steps 2 and 4 to Kernel and Homomorphism, respectively.
HOMOMORPHISM
Let G and G’ be groups with operations ∗and ∗′. A homomorphism of G into G’ is a mapping ?????? of G
intoG’ such that for every a and b in G,
?????? �∗� =?????? � ∗
′
?????? � .
Examples: Determine whether the given map is a homomorphism.
1. Let ??????:�
∗
→�
∗
under addition given by ?????? � =�
2
.
Answer: ?????? �+� = �+�
2
=�
2
+2��+�
2
≠�
2
+�
2
=?????? � +??????(�). Thus ?????? is NOT a
homomorphism.
2. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =�
2
.
Answer: ?????? �� = ��
2
=�
2
�
2
=�
2
�
2
=?????? � ∙??????(�). Thus ?????? is a homomorphism.
3. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =−�.
Answer: ?????? �� =− �� =−��≠ −� (−�)=?????? � ∙??????(�). Thus ?????? is NOT
ahomomorphism.
Types of Homomorphisms:
1. Epimorphism – surjective homomorphism
2. Monomorphism – injective homomorphism
3. Isomorphism – bijective homomorphism
4. Automorphism – isomorphism, domain and codomain are the same group
5. Endomorphism – homomorphism, domain and codomain are the same group
KERNEL
Let ??????:??????→??????′ be a homomorphism of groups. The subgroup ??????
−1
??????
′
= � ?????? ?????? ?????? � =??????′ is the
kernel of ??????, denoted by Ker(??????).
Examples: Find the kernel of the following homomorphism:
1. Let ??????:�
∗
→�
∗
under multiplication given by ?????? � =�
2
.
Answer: The identity element of the set of images is 1 under the operation of multiplication.
If �
2
=1 then �=−1 or �=1. Thus, Ker(??????)={−1,1}.
Page 2 of 3
2. Let ??????:�→� under addition given by ?????? � =5�.
Answer: The identity element of the set of images is 0 under the operation of addition. If 5x
= 0, then x = 0. Thus, Ker(??????)={0}.
COROLLARY:
A group homomorphism ??????:??????→??????′ is one – to – one map if and only if Ker(??????) = {??????}.
In view of this corollary, we can modify our steps in showing that two groups are isomorphic.
To show ??????:??????→??????′ is an isomorphism: (modified version)
1. Show ?????? is a homomorphism.
2. Show Ker(??????) = {??????}.
3. Show ?????? maps G onto G’.
Examples: (On showing isomorphism between G and G’)
A. Let us show that the binary structure <R, +> with operation the usual addition is isomorphic to
the structure <R
+, ∙> where ∙ is the usual multiplication.
Step 1. We have to somehow convert an operation of addition to multiplication. Recall from
�
�+�
= �
�
(�
�
) that the addition of exponents corresponds to multiplication of two quantities.
Thus we try defining ??????:�→�
+
by ?????? � =??????
�
for �∈�. Note that ??????
�
>0 for all �∈� so indeed
?????? � ∈�
+
.
Step 2. If ?????? � =??????(�), then ??????
�
=??????
�
. Taking the natural logarithm, we see that �=�, so ?????? is
indeed 1-1.
Step 3. If ??????∈�
+
, then ln(??????)∈� and ?????? ln?????? =??????
ln ??????
=??????. Thus ?????? is onto R
+.
Step 4.For�,�∈�, we have?????? �+� =??????
�+�
=??????
�
∙??????
�
=?????? � ∙?????? � .
Thus, ?????? is an isomorphism.
B. Let 2�= 2� �∈� , so that 2Z is the set of all even integers, positive, negative and zero. We
claim that <Z, +> is isomorphic to <2Z, +> where + is the usual addition.
Step 1. Define ??????:�→2� by ?????? � =2� for �∈�.
Step 2. If ?????? � =??????(�), then 2�=2� so �=�. Thus ?????? is 1-1.
Step 3. If �∈2�, then n is even so �=2� for �=�/2∈�. Hence ?????? � =2(�/2)=� so ?????? is onto
2Z.
Step 4. For�,�∈�, we have?????? �+� =2 �+� =2�+2�=?????? � +??????(�).
Thus, ?????? is an isomorphism.
C. Determine whether the given map ?????? is an isomorphism of the first binary structure with the
second.
1. <Z, +> with <Z, +> where ?????? � =2� for �∈�.
Answer: ?????? is 1-1, but NOT onto. Thus, ?????? is NOT an isomorphism.
2. <Z, +> with <Z, +> where ?????? � =�+1 for �∈�.
Answer:?????? is 1-1, onto, but NOT operation preserving. Thus, ?????? is NOT an isomorphism.
2. Let ??????:�→� under addition given by ?????? � =5�.
Answer: The identity element of the set of images is 0 under the operation of addition. If 5x
= 0, then x = 0. Thus, Ker(??????)={0}.
COROLLARY:
A group homomorphism ??????:??????→??????′ is one – to – one map if and only if Ker(??????) = {??????}.
In view of this corollary, we can modify our steps in showing that two groups are isomorphic.
To show ??????:??????→??????′ is an isomorphism: (modified version)
1. Show ?????? is a homomorphism.
2. Show Ker(??????) = {??????}.
3. Show ?????? maps G onto G’.
Examples: (On showing isomorphism between G and G’)
A. Let us show that the binary structure <R, +> with operation the usual addition is isomorphic to
the structure <R
+, ∙> where ∙ is the usual multiplication.
Step 1. We have to somehow convert an operation of addition to multiplication. Recall from
�
�+�
= �
�
(�
�
) that the addition of exponents corresponds to multiplication of two quantities.
Thus we try defining ??????:�→�
+
by ?????? � =??????
�
for �∈�. Note that ??????
�
>0 for all �∈� so indeed
?????? � ∈�
+
.
Step 2. If ?????? � =??????(�), then ??????
�
=??????
�
. Taking the natural logarithm, we see that �=�, so ?????? is
indeed 1-1.
Step 3. If ??????∈�
+
, then ln(??????)∈� and ?????? ln?????? =??????
ln ??????
=??????. Thus ?????? is onto R
+.
Step 4.For�,�∈�, we have?????? �+� =??????
�+�
=??????
�
∙??????
�
=?????? � ∙?????? � .
Thus, ?????? is an isomorphism.
B. Let 2�= 2� �∈� , so that 2Z is the set of all even integers, positive, negative and zero. We
claim that <Z, +> is isomorphic to <2Z, +> where + is the usual addition.
Step 1. Define ??????:�→2� by ?????? � =2� for �∈�.
Step 2. If ?????? � =??????(�), then 2�=2� so �=�. Thus ?????? is 1-1.
Step 3. If �∈2�, then n is even so �=2� for �=�/2∈�. Hence ?????? � =2(�/2)=� so ?????? is onto
2Z.
Step 4. For�,�∈�, we have?????? �+� =2 �+� =2�+2�=?????? � +??????(�).
Thus, ?????? is an isomorphism.
C. Determine whether the given map ?????? is an isomorphism of the first binary structure with the
second.
1. <Z, +> with <Z, +> where ?????? � =2� for �∈�.
Answer: ?????? is 1-1, but NOT onto. Thus, ?????? is NOT an isomorphism.
2. <Z, +> with <Z, +> where ?????? � =�+1 for �∈�.
Answer:?????? is 1-1, onto, but NOT operation preserving. Thus, ?????? is NOT an isomorphism.
Page 3 of 3
3. <R, ∙> with <R, ∙> where ?????? � =�
3
for �∈�.
Answer:?????? is 1-1, onto, and operation preserving. Thus, ?????? is an isomorphism.
4. <R, +> with <R
+, ∙> where ?????? ?????? =0.5
??????
for ??????∈�.
Answer:?????? is 1-1, onto, and operation preserving. Thus, ?????? is an isomorphism.
CAYLEY’S THEOREM
Every group is isomorphic to a group of permutations.
Example:
Let ??????={2,4,6,8}⊆Z10, where G forms a group with respect to multiplication modulo 10. Write out
the elements of a group of permutations that are isomorphic to G, and exhibit an isomorphism from
G to this group.
Solution:
Let ??????
�:??????→??????′ be defined by ??????
� � =�� for each �∈ G. Then we have the following permutations:
??????
2=
??????
2 2 =4
??????
2 4 =8
??????
2 6 =2
??????
2 8 =6
??????
4=
??????
4 2 =8
??????
4 4 =6
??????
4 6 =4
??????
4 8 =2
??????
6=
??????
6 2 =2
??????
6 4 =4
??????
6 6 =6
??????
6 8 =8
??????
8=
??????
8 2 =6
??????
8 4 =2
??????
8 6 =8
??????
8 8 =4
Thus, the set ??????
′
={??????
2,??????
4,??????
6,??????
8} is a group of permutations and the mapping ??????:??????→??????′ is defined
by
??????:
?????? 2 =??????
2
?????? 4 =??????
4
?????? 6 =??????
6
?????? 8 =??????
8
is an isomorphism from G to G’.
CAYLEY TABLE
A Cayley table (or operation table) of a (finite) group is a table with rows and columns labelled by
the elements of the group and the entry ??????∗ℎ in the row labelled g and column labelled h.
CAYLEY DIGRAPHS (DIRECTED GRAPHS)/CAYLEY DIAGRAMS
For each generating set S of a finite group G, there is a directed graph representing the
group in terms of the generators of S.
A digraph consists of a finite number of points, called vertices of the digraph, and some arcs
(each with a direction denoted by an arrowhead) joining vertices.
In a digraph for a group G using the generator set S we have one vertex, represented by a
dot, for each element of G.
Each generator in S is denoted by one type of arc.
Example:
At the right is a possible digraph for�
6 with�={2,3} using for
<2> and --- for <3>.
3. <R, ∙> with <R, ∙> where ?????? � =�
3
for �∈�.
Answer:?????? is 1-1, onto, and operation preserving. Thus, ?????? is an isomorphism.
4. <R, +> with <R
+, ∙> where ?????? ?????? =0.5
??????
for ??????∈�.
Answer:?????? is 1-1, onto, and operation preserving. Thus, ?????? is an isomorphism.
CAYLEY’S THEOREM
Every group is isomorphic to a group of permutations.
Example:
Let ??????={2,4,6,8}⊆Z10, where G forms a group with respect to multiplication modulo 10. Write out
the elements of a group of permutations that are isomorphic to G, and exhibit an isomorphism from
G to this group.
Solution:
Let ??????
�:??????→??????′ be defined by ??????
� � =�� for each �∈ G. Then we have the following permutations:
??????
2=
??????
2 2 =4
??????
2 4 =8
??????
2 6 =2
??????
2 8 =6
??????
4=
??????
4 2 =8
??????
4 4 =6
??????
4 6 =4
??????
4 8 =2
??????
6=
??????
6 2 =2
??????
6 4 =4
??????
6 6 =6
??????
6 8 =8
??????
8=
??????
8 2 =6
??????
8 4 =2
??????
8 6 =8
??????
8 8 =4
Thus, the set ??????
′
={??????
2,??????
4,??????
6,??????
8} is a group of permutations and the mapping ??????:??????→??????′ is defined
by
??????:
?????? 2 =??????
2
?????? 4 =??????
4
?????? 6 =??????
6
?????? 8 =??????
8
is an isomorphism from G to G’.
CAYLEY TABLE
A Cayley table (or operation table) of a (finite) group is a table with rows and columns labelled by
the elements of the group and the entry ??????∗ℎ in the row labelled g and column labelled h.
CAYLEY DIGRAPHS (DIRECTED GRAPHS)/CAYLEY DIAGRAMS
For each generating set S of a finite group G, there is a directed graph representing the
group in terms of the generators of S.
A digraph consists of a finite number of points, called vertices of the digraph, and some arcs
(each with a direction denoted by an arrowhead) joining vertices.
In a digraph for a group G using the generator set S we have one vertex, represented by a
dot, for each element of G.
Each generator in S is denoted by one type of arc.
Example:
At the right is a possible digraph for�
6 with�={2,3} using for
<2> and --- for <3>.
Page 4 of 3
University of Santo Tomas
College of Education
España, Manila
MATH 115: Abstract Algebra
Performance Task 1: Oral Presentation
Written Report on
Isomorphism
Submitted by:
CAPIOSO, RIKKI JOI A.
CHUA, ERNESTO ERIC III A.
GONZALES, MA. IRENE G.
PARK, MIN YOUNG
4MM
Submitted to:
Assoc. Prof. JOEL L. ADAMOS
Course Facilitator
Date Submitted:
08 April 2015, Wednesday
University of Santo Tomas
College of Education
España, Manila
MATH 115: Abstract Algebra
Performance Task 1: Oral Presentation
Written Report on
Isomorphism
Submitted by:
CAPIOSO, RIKKI JOI A.
CHUA, ERNESTO ERIC III A.
GONZALES, MA. IRENE G.
PARK, MIN YOUNG
4MM
Submitted to:
Assoc. Prof. JOEL L. ADAMOS
Course Facilitator
Date Submitted:
08 April 2015, Wednesday
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