Each of the following is a set of four groups. In each set, determine which grou
ID: 3113867 • Letter: E
Question
Each of the following is a set of four groups. In each set, determine which groups are isomorphic to which others. Prove your answers, and use Exercise A3 where convenient. 1 Z_4 Z_2 times Z_2 P_2 V [P_2 denotes the group of subsets of a two-element set. (See Chapter 3, Exercise C.) V denotes the group of the four complex numbers {i, -i, 1, -1} with respect to multiplication.] 2 S_3 Z_6 Z_3 times Z_2 Z*_7 (Z*_7 denotes the group {1, 2, 3, 4, 5, 6} with multiplication modulo 7. The product modulo 7 of a and b is the remainder of ab after division by 7.) 3 Z_8 P_3 Z_2 times Z_2 times Z_2 D_4 (D_4 is the group of symmetries of the square.) 4The groups having the following Cayley diagrams:Explanation / Answer
Z4
+
[ 0]
[ 1]
[ 2]
[ 3]
[ 0]
[ 1]
[ 2]
[ 3]
[ 0]
[ 1]
[ 1]
[ 2]
[ 3]
[ 0]
[ 2]
[ 2]
[ 3]
[ 0]
[ 1]
[ 3]
[ 3]
[ 0]
[ 1]
[ 2]
Z2 x Z2
+
([ 0] , [ 0] )
([ 0] , [ 1] )
([ 1] , [ 0] )
([ 1] , [ 1] )
([ 0] , [ 0] )
([ 0] , [ 0] )
([ 0] , [ 1] )
([ 1] , [ 0] )
([ 1] , [ 1] )
([ 0] , [ 1] )
([ 0] , [ 1] )
([ 0] , [ 0] )
([ 1] , [ 1] )
([ 1] , [ 0] )
([ 1] , [ 0] )
([ 1] , [ 0] )
([ 1] , [ 1] )
([ 0] , [ 0] )
([ 0] , [ 1] )
([ 1] , [ 1] )
([ 1] , [ 1] )
([ 1] , [ 0] )
([ 0] , [ 1] )
([ 0] , [ 0] )
P2
*
Æ
{ 1}
{ 2}
{ 1,2}
Æ
Æ
{ 1}
{ 2}
{ 1,2}
{ 1}
{ 1}
Æ
{ 1,2}
{ 2}
{ 2}
{ 2}
{ 1,2}
Æ
{ 1}
{ 1,2}
{ 1,2}
{ 2}
{ 1}
Æ
V
x
i
-i
1
-1
i
-1
1
i
-i
-i
1
-1
-i
i
1
i
-i
1
-1
-1
-i
i
-1
1
Consider Z4 and Z2 x Z2
Z2 x Z2 has ( [ 0] ,[ 0] ) is identity and every element in
Z2 x Z2 has own inverse
Z4 has [ 0] is identity but only one element that is
and [ 2] have own inverse
Consider Z4 and P2
Z4 has [ 2] have own inverse
but P2 has Æ is identity and every element in P2 has own inverse
Then Z4 is non-isomorphic with P2
Consider Z4 and V
Z4 has [ 0] is identity and has one element that is
[ 2] is own inverse and [ 1] ,[ 3] are inverse of each other
V has 1 is identity and has one element that is-1 is own inverse and i,-i are inverse of each other too
Let f : Z4® V and
f ([ 0] ) = 1
f ( [ 1] ) = i
f ( [ 2] ) = -1
f ( [ 3] ) = -i
Then Z4 @ V
Consider Z2 x Z2 and P2
Z2 x Z2 has ([ 0] ,[ 0] ) is identity
and every element are own inverse
P2 has Æ is identity
and every element are own inverse
Let g : Z2 x Z2® P2
and g([ 0] , [ 0] ) = Æ
g([ 0] , [ 1] ) = { 1}
g([ 1] , [ 0] ) = { 2}
g([ 1] , [ 1] ) = { 1,2}
Then Z2 x Z2 @ P2
Consider Z2 x Z2 and V
Z2 x Z2 has every element have own inverse
but V has one elements that are -1 are own inverse
Then Z2 x Z2 is non-isomorphic with V
Consider P2 and V
P2 has every elements are own inverse
but V has only one element that is -1 has own inverse
Then P2 is non-isomorphic with V
2)
S3
*
(1)
(12)
(13)
(23)
(123)
(132)
(1)
(1)
(12)
(13)
(23)
(123)
(132)
(12)
(12)
(1)
(123)
(132)
(13)
(23)
(13)
(13)
(132)
(1)
(123)
(23)
(12)
(23)
(23)
(123)
(132)
(1)
(12)
(13)
(123)
(123)
(23)
(12)
(13)
(132)
(1)
(132)
(132)
(13)
(23)
(12)
(1)
(123)
Z6
+
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 0]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 1]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 0]
[ 2]
[ 2]
[ 3]
[ 4]
[ 5]
[ 0]
[ 1]
[ 3]
[ 3]
[ 4]
[ 5]
[ 0]
[ 1]
[ 2]
[ 4]
[ 4]
[ 5]
[ 0]
[ 1]
[ 2]
[ 3]
[ 5]
[ 5]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
Z3 x Z2
+
([ 0] ,[ 0])
([ 0] ,[ 1])
([ 1] ,[ 0])
([ 1] ,[ 1] )
([ 2] ,[ 0] )
([ 2] ,[ 1] )
([ 0] ,[ 0] )
([ 0] ,[ 0])
([ 0] ,[ 1])
([ 1] ,[ 0])
([ 1] ,[ 1] )
([ 2] ,[ 0] )
([ 2] ,[ 1] )
([ 0] ,[ 1] )
([ 0] ,[ 1])
([ 0] ,[ 0])
([ 1] ,[ 1])
([ 1] ,[ 0] )
([ 2] ,[ 1] )
([ 2] ,[ 0] )
([ 1] ,[ 0] )
([ 1] ,[ 0])
([ 1] ,[ 1])
([ 2] ,[ 0])
([ 2] ,[ 1] )
([ 0] ,[ 0] )
([ 0] ,[ 1] )
([ 1] ,[ 1] )
([ 1] ,[ 1])
([ 1] ,[ 0])
([ 2] ,[ 1])
([ 2] ,[ 0] )
([ 0] ,[ 1] )
([ 0] ,[ 0] )
([ 2] ,[ 0] )
([ 2] ,[ 0])
([ 2] ,[ 1])
([ 0] ,[ 0])
([ 0] ,[ 1] )
([ 1] ,[ 0] )
([ 1] ,[ 1] )
([ 2] ,[ 1] )
([ 2] ,[ 1])
([ 2] ,[ 0])
([ 0] ,[ 1])
([ 0] ,[ 0] )
([ 1] ,[ 1] )
([ 1] ,[ 0] )
Z7*
*
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
1
3
5
3
3
6
2
5
1
4
4
4
1
5
2
6
3
5
5
3
1
6
4
2
6
6
5
4
3
2
1
Consider S3 has (1) is identity
and (12) ,(13) ,(23) are own inverse
and (123) is inverse of (132)
Z6 has [ 0] is identity
and [ 3] is own inverse
and [ 1] is inverse of [ 5] , [ 2] is inverse [ 4]
Z3 x Z2 has ( [ 0] , [ 0] ) is identity
and ( [ 0] , [ 1] ) is own inverse
and ( [ 1] , [ 0] ) is inverse of ( [ 2] , [ 0] ) ,
( [ 1] , [ 1] ) is inverse of ( [ 2] , [ 1] )
Z7* has 1 is identity
and 6 is own inverse
and 2 is inverse of 4 , 3 is inverse of 5
From that , can conclude :
S3 is non-isomorphic with Z6
S3 is non-isomorphic with Z3 x Z2
S3 is non-isomorphic with Z7*
Z6 is non-isomorphic with Z7*
Z3 x Z2 is non-isomorphic with Z7*
Consider g : Z6@ Z3 x Z2
g( [ 0] ) = ( [ 0] , [ 0] )
g( [ 3] ) = ( [ 0] , [ 1] )
g( [ 1] ) = ( [ 1] , [ 0] )
g( [ 2] ) = ( [ 1] , [ 1] )
g( [ 4] ) = ( [ 2] , [ 1] )
g( [ 5] ) = ( [ 2] , [ 0] )
Z6 @ Z3 x Z2
3)
Z8
+
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 6]
[ 7]
[ 0]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 6]
[ 7]
[ 1]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 6]
[ 7]
[ 0]
[ 2]
[ 2]
[ 3]
[ 4]
[ 5]
[ 6]
[ 7]
[ 0]
[ 1]
[ 3]
[ 3]
[ 4]
[ 5]
[ 6]
[ 7]
[ 0]
[ 1]
[ 2]
[ 4]
[ 4]
[ 5]
[ 6]
[ 7]
[ 0]
[ 1]
[ 2]
[ 3]
[ 5]
[ 5]
[ 6]
[ 7]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 6]
[ 6]
[ 7]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 7]
[ 7]
[ 0]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5]
[ 6]
P3
*
Æ
{ 1}
{ 2}
{ 3}
{ 1,2}
{ 1,3}
{ 2,3}
{ 1,2,3}
Æ
Æ
{ 1}
{ 2}
{ 3}
{ 1,2}
{ 1,3}
{ 2,3}
{ 1,2,3}
{ 1}
{ 1}
Æ
{ 1,2}
{ 1,3}
{ 2}
{ 3}
{ 1,2,3}
{ 2,3}
{ 2}
{ 2}
{ 1,2}
Æ
{ 2,3}
{ 1}
{ 1,2,3}
{ 3}
{ 1,3}
{ 3}
{ 3}
{ 1,3}
{ 2,3}
Æ
{ 1,2,3}
{ 1}
{ 2}
{ 1,2}
{ 1,2}
{ 1,2}
{ 2}
{ 1}
{ 1,2,3}
Æ
{ 2,3}
{ 1,3}
{ 3}
{ 1,3}
{ 1,3}
{ 3}
{ 1,2,3}
{ 1}
{ 2,3}
Æ
{ 1,2}
{ 2}
{ 2,3}
{ 2,3}
{ 1,2,3}
{ 3}
{ 2}
{ 1,3}
{ 1,2}
Æ
{ 1}
{ 1,2,3}
{ 1,2,3}
{ 2,3}
{ 1,3}
{ 1,2}
{ 3}
{ 2}
{ 1}
Æ
(A*B) = (A-B)È (B-A)
Z2 x Z2 x Z2
+
([ 0] ,[ 0] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 0] )
([ 0] ,[ 0] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 1] )
([ 0] ,[ 0] ,[ 1])
([ 0] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 0] ,[ 1] ,[ 0] )
([ 0] ,[ 1] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 0])
([ 1] ,[ 1] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 0])
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 0] ,[ 0] )
([ 1] ,[ 0] ,[ 0])
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 0] ,[ 0] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 0] ,[ 1] ,[ 1] )
([ 0] ,[ 1] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 0])
([ 1] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 0] ,[ 0])
([ 1] ,[ 0] ,[ 1] )
([ 1] ,[ 0] ,[ 1])
([ 1] ,[ 0] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 0] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 1])
([ 0] ,[ 1] ,[ 0])
([ 1] ,[ 1] ,[ 0] )
([ 1] ,[ 1] ,[ 0])
([ 1] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 1])
([ 0] ,[ 0] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 1] ,[ 1] ,[ 1] )
([ 1] ,[ 1] ,[ 1])
([ 1] ,[ 1] ,[ 0])
([ 1] ,[ 0] ,[ 1])
([ 0] ,[ 1] ,[ 1])
([ 1] ,[ 0] ,[ 0])
([ 0] ,[ 1] ,[ 0])
([ 0] ,[ 0] ,[ 1])
([ 0] ,[ 0] ,[ 0])
D4
*
I
H
V
D1
D2
R1
R2
R3
I
I
H
V
D1
D2
R1
R2
R3
H
H
I
R2
R1
R3
D1
V
D2
V
V
R2
I
R3
R1
D2
H
D1
D1
D1
R3
R1
I
R2
V
D2
H
D2
D2
R1
R3
R2
I
H
D1
V
R1
R1
D2
D1
H
V
R2
R3
I
R2
R2
V
H
D2
D1
R3
I
R1
R3
R3
D1
D2
V
H
I
R1
R2
Consider Z8 has [ 0] is identity
and [ 4] is own inverse
and [ 1] is inverse of [ 7] , [ 2] is inverse of [ 6] ,
[ 3] is inverse of [ 5]
P3 has Æ is identity
and every elements are own inverse
Z2 x Z2 x Z2 has ( [ 0] , [ 0] , [ 0] ) is identity
and every elements are own inverse
D4 has I is identity
and H,V,D1,D2,R2 are own inverse
and R1 is inverse of R2
From that , can conclude:
Z8 is non-isomorphic with P2
Z8 is non-isomorphic with Z2 x Z2 x Z2
Z8 is non-isomorphic with D4
P3 is non-isomorphic with D4
D4 is non-isomorphic with Z2 x Z2 x Z2
Consider f : P3® Z2 x Z2 x Z2
f (Æ ) = ( [ 0] , [ 0] , [ 0] )
f ({ 1} ) = ( [ 0] , [ 0] , [ 1] )
f({ 2} ) = ( [ 0] , [ 1] , [ 0] )
f ({ 3} ) = ( [ 1] , [ 0] , [ 0] )
f ({ 1,2} ) = ( [ 0] , [ 1] , [ 1] )
f ({ 1,3} ) = ( [ 1] , [ 0] , [ 1] )
f ({ 2,3} ) = ( [ 1] , [ 1] , [ 0] )
f ({ 1,2,3} ) = ( [ 1] , [ 1] , [ 1] )
P3 =Z2 x Z2 x Z2
+
[ 0]
[ 1]
[ 2]
[ 3]
[ 0]
[ 1]
[ 2]
[ 3]
[ 0]
[ 1]
[ 1]
[ 2]
[ 3]
[ 0]
[ 2]
[ 2]
[ 3]
[ 0]
[ 1]
[ 3]
[ 3]
[ 0]
[ 1]
[ 2]
Related Questions
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