Download MA 407: Introduction to Modern Algebra Homework

MA 407: Introduction to Modern Algebra
Homework
Hoon Hong
1
Group Theory
1.1
Definition of Group
1. State the definition of group..
2. Is the following a group? If not, why not?
(a) ({0, 1, 2} , +)
(b) ({0, 1, 2} ,
a\b 0
0
0
1
1
2
2
})
1
1
1
0
where a } b is given by
2
2
0
1
(c) (Z∗ , +) where Z∗ = Z\ {0}
(d) (Q, ×).
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1.2
Example of Group: Zn
1. State the definition of the following notions:
(a) Zn
(b) +n
2. For (Z4 , +4 ) , do the followings:
(a) Construct the operation table.
(b) Check if it satisfies the closure property.
(c) Check if it satisfies the associative property.
(d) Check if it satisfies the identity property.
(e) Check if it satisfies the inverse property.
3. Find 27−1 in Z120 .
4. Find a formula for x−1 in Zn .
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1.3
Example of Group: Un
1. State the definition of the following notions:
(a) Un
(b) ×n
2. For (U8 , ×8 ) , do the followings:
(a) Construct the operation table.
(b) Check if it satisfies the closure property.
(c) Check if it satisfies the associative property.
(d) Check if it satisfies the identity property.
(e) Check if it satisfies the inverse property.
3. Find 197−1 in U3000 .
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1.4
Example of Group: Sn
1. State the definition of the following notions:
(a) permutation:
(b) Sn
(c) operation on Sn
2. For (S3 , ◦) , do the followings:
(a) Construct the operation table.
(b) Check if it satisfies the closure property.
(c) Check if it satisfies the associative property.
(d) Check if it satisfies the identity property.
(e) Check if it satisfies the inverse property.
3. Find
1
6
2
1
3
4
4
5
5
3
6
2
−1
.
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1.5
Example of Group: An
1. State the definition of the following notions:
(a) cycle
(b) transposition
(c) even, odd permutation
(d) An
2. For (A3 , ◦) , do the followings:
(a) Construct the operation table.
(b) Check if it satisfies the closure property.
(c) Check if it satisfies the associative property.
(d) Check if it satisfies the identity property.
(e) Check if it satisfies the inverse property.
3. Find at least three different ways to write
1
6
2 3
1 4
4
5
5
3
6
2
2 3
4 5
4
3
5
2
6
1
as a composition of transpositions.
4. Find at least three different ways to write
1
6
as a composition of transpositions.
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1.6
Example of Group: Dn
1. State the definition of the following notions:
(a) Dn
2. For (D4 , ◦) , do the followings:
(a) Construct the operation table.
(b) Check if it satisfies the closure property.
(c) Check if it satisfies the associative property.
(d) Check if it satisfies the identity property.
(e) Check if it satisfies the inverse property.
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1.7
Uniqueness of Identity
1. For each of the following groups
(a) (Z4 , +4 )
• Count the number of identity elements.
• Make some observations about them.
(b) (U8 , ×8 )
• Count the number of identity elements.
• Make some observations about them.
(c) (S3 , ◦)
• Count the number of identity elements.
• Make some observations about them.
(d) (A3 , ◦)
• Count the number of identity elements.
• Make some observations about them.
(e) (D4 , ◦)
• Count the number of identity elements.
• Make some observations about them.
2. Prove: Let G be a group. Then G has only one identity element.
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1.8
Uniqueness of Inverse
1. For each of the following groups
(a) (Z4 , +4 )
• Count the number of inverse elements for each element.
• Make some observations about them.
(b) (U8 , ×8 )
• Count the number of inverse elements for each element.
• Make some observations about them.
(c) (S3 , ◦)
• Count the number of inverse elements for each element.
• Make some observations about them.
(d) (A3 , ◦)
• Count the number of inverse elements for each element.
• Make some observations about them.
(e) (D4 , ◦)
• Count the number of inverse elements for each element.
• Make some observations about them.
2. Prove: Let G be a group. Then every element of G has only one inverse.
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1.9
Subgroup
1. State the definition of the following notions:
(a) subgroup (≤)
2. For each of the following groups, find all the subgroups.
(a) (Z4 , +4 )
(b) (U8 , ×8 )
(c) (S3 , ◦)
(d) (A3 , ◦)
(e) (D4 , ◦)
3. Prove: Let G be a group and H ⊆ G.We have H ≤ G if (1) H 6= ∅ and (2) ∀a, b ∈ H ab−1 ∈ H.
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1.10
Normal Subgroups and Quotient Groups
1. State the definition of the following notions
(a) normal subgroup
(b) quotient set
(c) operation on set
2. For each of the following groups G do
• Find all the normal subgroups of G.
• For each normal subgroup H, construct the operation table on G/H.
• Check if G/H is a group.
(a) (Z4 , +4 )
(b) (U8 , ×8 )
(c) (S3 , ◦)
(d) (A3 , ◦)
(e) (D4 , ◦)
3. Prove: Let G be a group and let H C G. Then the operation over G/H is well defined, that is, if
aH = a0 H and bH = b0 H then (ab) H = (a0 b0 ) H.
4. Prove: Let G be a group and let H C G. Then G/H is a group.
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1.11
Homomorphism, Image, Kernel and Isomorphism Theorem
1. State the definition of the following notions
(a) Homomorphism
(b) Isomorphism
(c) Isomorphic (∼
=)
(d) Kernel
(e) Image
2. For each of the following maps φ : (G, ◦) −→ (G0 , ◦0 ) do
• Draw the map diagram for φ.
• Construct the operation table for im φ.
• Construct the operation table for ker φ.
• Construct the operation table for G/ ker φ.
• Draw the map diagram for the “natural” isomorphism that shows G/ ker φ ∼
= im φ.
(a) φ : (Z9 , +9 ) −→ (Z9 , +9 ), given by x 7−→ 3 ×9 x
1 if x is 1 or 3
(b) φ : (U8 , ×8 ) −→ (U8 , ×8 ), given by x 7−→
5 otherwise
1 if x is an even permutation
.
(c) φ : (S3 , ◦) −→ (U8 , ×8 ), given by x 7−→
3 otherwise
0 if x is a rotation
(d) φ : (D4 , ◦) −→ (Z4 , +4 ), given by x 7−→
2 otherwise
3. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then φ (e) = e0 .
−1
4. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then ∀a ∈ G φ a−1 = φ (a) .
5. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then im φ ≤ G0 .
6. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then ker φ ≤ G.
7. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then ker φ C G.
8. Prove: Let φ : (G, ◦) −→ (G0 , ◦0 ) be a homomorphism. Then G/ ker φ ∼
= im φ.
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2
Ring Theory
2.1
Definition of Ring
1. State the definitions of the following abstract notions
(a) Ring
(b) Commutative Ring
(c) Ring with Unity
(d) Commutative Ring with Unity (CRU)
(e) Integral domain
(f) Field
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2.2
Examples of Ring
1. State the definitions of the following concrete notations.
(a) kZ
(b) Mn (S)
(c) S [x]
(d) S (x)
2. Classify the following algebraic structures, using a Venn diagram (as we have done in the class).
N
M2 (N)
N [x]
Z
M2 (Z)
Z [x]
Q
M2 (Q)
Q [x]
Q (x)
R
M2 (R)
R [x]
R (x)
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C
M2 (C)
C [x]
C (x)
Z3
M2 (Z3 )
Z3 [x]
Z3 (x)
Z6
M2 (Z6 )
Z6 [x]
3Z
M2 (3Z)
3Z [x]
2.3
Uniqueness of identity and inverse
1. Prove: Let R be a ring. Then there is only one additive identity.
2. Prove: Let R be a ring. Then every element of R has has only one additive inverse.
3. Prove: Let R be a ring with unity. Then there is only one multiplicative identity.
4. Prove: Let R be a field. Then every non-zero element of R has only one multiplicative inverse.
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2.4
Subring
1. State the definitions of the following notions:
(a) Subring
2. Check the truth of the followings.
(a) 3Z ≤ Z
(b) {0, 5} ≤ Z12
(c) 2Z12 ≤ Z12
(d) 3Z12 ≤ Z12
(e) 4Z12 ≤ Z12
(f) 6Z12 ≤ Z12
(g) {a + bi ∈ C : a, b ∈ Z} ≤ C
a 0
(h)
: a, b, c ∈ R ≤ M2 (R)
b c
3. Prove: Let R be a ring and S ⊆ R. We have S ≤ R if
(a) S 6= ∅
(b) ∀a, b ∈ S a + (−b) ∈ S
(c) ∀a, b ∈ S a ∙ b ∈ S
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2.5
Ideal and Quotient ring
1. State the definitions of the following notions:
(a) Ideal
(b) Generated set
(c) Quotient set
2. Check the truth of the followings.
3Z C Z
{0, 5} C Z12
2Z12 C Z12
3Z12 C Z12
4Z12 C Z12
6Z12 C Z12
Z [i] = {a + bi ∈ C : a, b ∈ Z} C C
a 0
(h)
: a, b, c ∈ R C M2 (R)
b c
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3. List the elements of the following sets:
(a) h3i as an ideal of Z
(b) h8, 12i as an ideal of Z
(c) h2i as an ideal of Z8
(d) h4i as an ideal of Z8
(e) hxi as an ideal of Z2 [x]
(f) x2 as an ideal of Z2 [x]
4. For each of the following structures
• List the elements.
• Construct the operation tables for addition and multiplication.
• Verify that it is a ring.
(a)
(b)
(c)
(d)
(e)
Z/ h3i
Z8 / h2i
Z8 / h4i
Z2 [x] / hxi
Z2 [x] / x2
5. Prove: Let R be a CRU and let a1 , . . . , an ∈ R. Then ha1 , . . . , an i C R.
6. Prove: Let R be a ring and let I C R. Then the addition operation on R/I is well defined.
7. Prove: Let R be a ring and let I C R. Then the multiplication operation on R/I is well defined.
8. Prove: Let R be a ring and let I C R. Then R/I is a ring.
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2.6
Homomorphism, Isomorphism, Image and Kernel
1. State the definition of the following notions
(a) Homomorphism
(b) Isomorphism
(c) Isomorphic (∼
=)
(d) Kernel
(e) Image
2. For each of the following maps φ : (R, +, ∙) −→ (R0 , +0 , ∙0 ) do
• Draw the map diagram for φ.
• Construct the operation table for im φ.
• Construct the operation table for ker φ.
• Construct the operation table for G/ ker φ.
• Draw the map diagram for the “natural” isomorphism that shows G/ ker φ ∼
= im φ.
(a) φ : Z −→ Z5 , given by x 7−→ x mod 5
(b) φ : Z4 −→ Z10 , given by x 7−→ (5x) mod 10
(c) φ : Z5 −→ Z10 , given by x 7−→ (6x) mod 10
3. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then φ (0) = 00 .
4. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then φ (−a) =
−0 φ (a).
5. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then im φ ≤ R0 .
6. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then ker φ ≤ R.
7. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then ker φ C R.
8. Prove: Let (R, +, ∙) and (R0 , +0 , ∙0 ) be rings. Let φ : R −→ R0 be a homomorphism. Then R/ ker φ ∼
=
im φ.
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