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

3/19/2019

2

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

3/19/2019

3

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

3/19/2019

4

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 ,

3/19/2019

5

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

3/19/2019

6

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

3/19/2019

7

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

3/19/2019

8

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?

3/19/2019

9

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

3/19/2019

10

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

3/19/2019

11

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.

3/19/2019

12

Practice

Find all abelian groups, up to isomorphism, of order 180.

Vocabulary

Decomposable versus indecomposable

3/19/2019

13

The End of Section 11