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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
๐ =G, cyclic of order n. bG (b=as for some s) generates a cyclic subgroup ๐ of order โwhere d = gcd(n,s).
8 ๐คโ๐๐๐ 8 โค
Today
โข Subgroups of cyclic groupsโข Automorphismsโข Permutations in Algebraโข Challenges of Permutationsโข Permutation notationโข Computing function values with a permutationโข Finding inversesโข Composing permutationsโข A Cayley table for a group of permutations
Donโt Forget
โข Pick up today
โ In class exercises:Practice ยง8a
โข Due Monday
โ Reading Quiz 8 (on Bb, due before class)
โข Due tomorrow
โ Problem Set 5 (due 4 PM)
6. Find the number of elements of
โข the cyclic subgroup of โค generated by 15
โข the cyclic subgroup ๐ of โโ,ยท
7. Find the number of automorphisms
โคโค
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8. Can you tell whether or notโฆ
6 ๐๐ ๐ ๐๐๐๐๐๐๐ก๐๐ ๐๐ โค ?4 ๐๐ ๐ ๐๐๐๐๐๐๐ก๐๐ ๐๐ โค ?
9. Find the order of
the cyclic subgroup of โค generated by 15
The End of Section 6
Definition
A permutation of a set A is a bijection๐: ๐ด โ ๐ด
Example:๐: 1, 2, 3 โ 1, 2, 3
๐: 1 โฆ 2 2 โฆ 3 3 โฆ 1
Challenges of Working with Permutations in Algebra
1. We ainโt 8P6
2. Our objects are functions.
3. Functions act from right to left.
Examples
๐ 15
24
33
42
51
๐ 12
23
34
45
51
๐ 12
21
34
43
55
๐พ 13
22
35
44
51
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Return to orbits
๐ 15
24
33
42
51
๐ 12
23
34
45
51
๐ 12
21
34
43
55
๐พ 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
โข Proof I promised
โข Cayleyโs Theorem
โข Orbits
โข Cycles
โข Transpositions
Donโt Forget
โข Pick up today
โ In class exercises:Practice ยง8b
โข Due Wednesday
โ Reading Quiz 9 (on Bb, due before class)
โข Due Thursday
โ Problem Set 6
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Proof
p. 58 #49G a finite group with gGโ๐ โ โค such thatgn = e
Lemma 8.15
๐: ๐บ โถ ๐บ ๐ 1 1 โ๐๐๐๐๐๐๐โ๐๐ ๐ โ๐ ๐บ ๐บโฒ
Recall
3
1 2
D3
Cayleyโs Theorem
Every group is isomorphic to a group of permutations.
0 1 2 1 2 3
0
1
2
1
2
3
Recall
๐ 15
24
33
42
51
๐ 12
23
34
45
51
๐ 12
21
34
43
55
๐พ 13
22
35
44
51
1. Find ๐ช ,
2. Find ๐ช ,
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Donโt Open Your Book!!!
๐ฟ๐๐ก ๐, ๐ โ ๐ด, ๐ โ ๐๐~๐ ๐๐๐ ๐ ๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐ โ โคProve ~ 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
โข Orbits
โข Cycles
โข Transpositions
โข Alternating Group
โข Matrices and permutations
โข Another Equivalence relation
Donโt Forget
โข Pick up today
โ In class exercises:Practice ยง9
โข Due Monday
โ Reading Quiz 10 (on Bb, due before class)
โข Due Tomorrow
โ Problem Set 6
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Remember?
๐ฟ๐๐ก ๐, ๐ โ ๐ด, ๐ โ ๐๐~๐ ๐๐๐ ๐ ๐ ๐ ๐๐๐ ๐ ๐๐๐ ๐ โ โคProve ~ is an equivalence relation.
Do you remember?
An equivalence relation always gives rise to a ? .
Find all orbits
๐ 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 happensโฆ
d
c
b
a
1000
0100
0010
0001
Hint
p. 86 #46 (PS 7)๐ ๐ ๐ โฆ
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?
โข ๐ขโข ๐ขโข ๐ขโข ๐ข
Theorem
If H G, then ~ is an equivalence relation on G:
a~b a-1bH
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Today
โข An Equivalence Relation
โข Cosets
Donโt Forget
โข Pick up today
โ In class exercises:Practice ยง10
โข Due Thursday
โ Problem Set 7
โข 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 byโฆ
โคThe 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
โข Direct Products โข Fundamental Theorem of Finitely
Generated Abelian Groups
Donโt Forget
โข Pick up todayโ In class exercises:Practice ยง11
โ graded PS 6
โ Midterm Study questions
โข Due tomorrowโ Problem Set 7
โข No more Reading Quizzes until after midterm
1. Example
A , , B , , ,
A B
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1. Practice
List the elements of โค โค
2. Practice
a. Find the order of (3, 5) in โค โคb. Find the order of (1, 2) in โค โค
Theorem 11.5
The group โค โค is cyclic and is isomorphic to โค iff m and n are relatively prime.
Question
Is โค โค โค โ โค ?
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