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3/19/2019
1
Reflection
Give praise to the Lord, proclaim his name; make known among the nations what He has done.
1 Chronicles 16:8
Recall Theorem 6.14
𝑎 =G, cyclic of order n. bG (b=as for some s) generates a cyclic subgroup 𝑏 of order ⁄where d = gcd(n,s).
8 𝑤ℎ𝑒𝑟𝑒 8 ℤ
Today
• Subgroups of cyclic groups• Automorphisms• Permutations in Algebra• Challenges of Permutations• Permutation notation• Computing function values with a permutation• Finding inverses• Composing permutations• A Cayley table for a group of permutations
Don’t Forget
• Pick up today
– In class exercises:Practice §8a
• Due Monday
– Reading Quiz 8 (on Bb, due before class)
• Due tomorrow
– Problem Set 5 (due 4 PM)
6. Find the number of elements of
• the cyclic subgroup of ℤ generated by 15
• the cyclic subgroup 𝑖 of ℂ∗,·
7. Find the number of automorphisms
ℤℤ
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8. Can you tell whether or not…
6 𝑖𝑠 𝑎 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 𝑜𝑓 ℤ ?4 𝑖𝑠 𝑎 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 𝑜𝑓 ℤ ?
9. Find the order of
the cyclic subgroup of ℤ generated by 15
The End of Section 6
Definition
A permutation of a set A is a bijection𝜑: 𝐴 → 𝐴
Example:𝜇: 1, 2, 3 → 1, 2, 3
𝜇: 1 ↦ 2 2 ↦ 3 3 ↦ 1
Challenges of Working with Permutations in Algebra
1. We ain’t 8P6
2. Our objects are functions.
3. Functions act from right to left.
Examples
𝜎 15
24
33
42
51
𝜏 12
23
34
45
51
𝜌 12
21
34
43
55
𝛾 13
22
35
44
51
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Return to orbits
𝜎 15
24
33
42
51
𝜏 12
23
34
45
51
𝜌 12
21
34
43
55
𝛾 13
22
35
44
51
The End of Section 8a
Reflection:Fear the LORD, you his saints, for those who fear him lack nothing. Psalm 34:9
Challenges of Working with Permutations in Algebra
1.
2.
3.
Today
• Proof I promised
• Cayley’s Theorem
• Orbits
• Cycles
• Transpositions
Don’t Forget
• Pick up today
– In class exercises:Practice §8b
• Due Wednesday
– Reading Quiz 9 (on Bb, due before class)
• Due Thursday
– Problem Set 6
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Proof
p. 58 #49G a finite group with gG∃𝑛 ∈ ℤ such thatgn = e
Lemma 8.15
𝜑: 𝐺 ⟶ 𝐺 𝑎 1 1 ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚 ⇒𝜑 𝐺 𝐺′
Recall
3
1 2
D3
Cayley’s Theorem
Every group is isomorphic to a group of permutations.
0 1 2 1 2 3
0
1
2
1
2
3
Recall
𝜎 15
24
33
42
51
𝜏 12
23
34
45
51
𝜌 12
21
34
43
55
𝛾 13
22
35
44
51
1. Find 𝒪 ,
2. Find 𝒪 ,
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Don’t Open Your Book!!!
𝐿𝑒𝑡 𝑎, 𝑏 ∈ 𝐴, 𝜎 ∈ 𝑆𝑎~𝑏 𝑖𝑓𝑓 𝑏 𝜎 𝑎 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛 ∈ ℤProve ~ is an equivalence relation.
The End of Section 8b
Reflection:When you pass through the waters,
I will be with you;and when you pass through the rivers,
they will not sweep over you.When you walk through the fire,
you will not be burned;the flames will not set you ablaze.
Isaiah 43:2
Today
• Orbits
• Cycles
• Transpositions
• Alternating Group
• Matrices and permutations
• Another Equivalence relation
Don’t Forget
• Pick up today
– In class exercises:Practice §9
• Due Monday
– Reading Quiz 10 (on Bb, due before class)
• Due Tomorrow
– Problem Set 6
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Remember?
𝐿𝑒𝑡 𝑎, 𝑏 ∈ 𝐴, 𝜎 ∈ 𝑆𝑎~𝑏 𝑖𝑓𝑓 𝑏 𝜎 𝑎 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛 ∈ ℤProve ~ is an equivalence relation.
Do you remember?
An equivalence relation always gives rise to a ? .
Find all orbits
𝜌 12
21
34
43
55
Compute the indicated product
(2, 4, 6)(2, 3)(1, 5, 4)
Express as a product of transpositions
Algorithm at the bottom of page 90.
15
24
31
46
53
62
Even or odd?
What is the order of
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What happens…
d
c
b
a
1000
0100
0010
0001
Hint
p. 86 #46 (PS 7)𝑆 𝑆 𝑆 …
The End of Section 9
Reflection:When pride comes, then comes disgrace,
but with humility comes wisdom. Proverbs 11:2
What are the Group Axioms?
• 𝒢• 𝒢• 𝒢• 𝒢
Theorem
If H G, then ~ is an equivalence relation on G:
a~b a-1bH
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Today
• An Equivalence Relation
• Cosets
Don’t Forget
• Pick up today
– In class exercises:Practice §10
• Due Thursday
– Problem Set 7
• No more Reading Quizzes until after midterm
What to call the cells of the partition?
Left cosets gHRight cosets Hg
GH
S3
0 0 0 1 2 3
0
1
2
1
2
3
Question
What is |S3|? |H|?How many cosets were there?
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V
e a b c
e
a
b
c
Theorem of Lagrange
|G| finite and H G
Theorem of Lagrange
|G| finite and H G
Corollary: |G| = p, a prime
Contemplate
3 ____ = 15
e a b c
e e a b c
a a e c b
b b c e a
c c b a e
What exactly do we mean by…
ℤThe End of Section 10
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START Reflection:I sought the LORD, and he answered me; he delivered me from all my fears. Psalm 34:4
{0, 1, 2} = H S3
0 1 2 1 2 3
0 0 1 2 1 2 3
1 1 2 0 3 1 2
2 2 0 1 2 3 1
1 1 2 3 0 1 2
2 2 3 1 2 0 1
3 3 1 2 1 2 0
Today
• Direct Products • Fundamental Theorem of Finitely
Generated Abelian Groups
Don’t Forget
• Pick up today– In class exercises:Practice §11
– graded PS 6
– Midterm Study questions
• Due tomorrow– Problem Set 7
• No more Reading Quizzes until after midterm
1. Example
A , , B , , ,
A B
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1. Practice
List the elements of ℤ ℤ
2. Practice
a. Find the order of (3, 5) in ℤ ℤb. Find the order of (1, 2) in ℤ ℤ
Theorem 11.5
The group ℤ ℤ is cyclic and is isomorphic to ℤ iff m and n are relatively prime.
Question
Is ℤ ℤ ℤ ≃ ℤ ?
Theorem 11.12
The Fundamental Theorem of Finitely Generated Abelian GroupsEvery finitely generated abelian group is isomorphic to a direct product of cyclic groups of the form
Example
Find all abelian groups, up to isomorphism, of order 100.
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Practice
Find all abelian groups, up to isomorphism, of order 180.
Vocabulary
Decomposable versus indecomposable
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The End of Section 11
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