Modern Algebra 1 - City University of New York...Last time: Modular arithmetic. For any two integers...

Modern Algebra 1

Professor: Zajj DaughertyOffice hours: Wed 12:45–1:45 or by appointment in NAC 6/301.http://zdaugherty.ccnysites.cuny.edu/teaching/mA4900f17/

Last time:Chapter 0 review.If you missed, be sure to read over the 16-page handout from lasttime.

Last time: Modular arithmetic.

For any two integers a, n ∈ Z with n 6= 0, there are unique q, r ∈ Zwith

a = qn + r and 0 ≤ r < |n|.

We say two numbers a and b are congruent modulo (mod) n,written

a ≡ b (mod n) or a ≡n b

if they have the same remainder when divided by n.

−24 −18 −12 −6 0 6 12 18 24

−26 −20 −14 −8 −2 4 10 16 22 28

allcongr. to 4

mod 6



a = qn + r and 0 ≤ r < |n|.




−24 −18 −12 −6 0 6 12 18 24

−26 −20 −14 −8 −2 4 10 16 22 28

allcongr. to 4

mod 6



a = qn + r and 0 ≤ r < |n|.




−24 −18 −12 −6 0 6 12 18 24

−26 −20 −14 −8 −2 4 10 16 22 28

allcongr. to 4

mod 6

Last time: Modular arithmetic.For n ∈ Z>1,

a ∼ b if and only if a ≡ b (mod n)is an equivalence relation. The equiv class of a ∈ Z is called acongruence or residue class (mod n), and is given by

a = {a + kn | k ∈ Z} = {. . . , a− 2n, a− n, a, a + n, a + 2n, . . . }.−4n −3n −2n 2n 3n 4n0 n−n

aa−na−2n a+n a+2n a+3n a+4n a+5n

We call the set of congruence classes

Z/nZ = {0, 1, 2, . . . , n− 1}, read “Z mod n Z”.

Theorem. Fix n ∈ Z>0. Then

addition: a + b := a + b, andmultiplication: a ∗ b := a ∗ b

are well-defined binary operations on the set of congruence classesmodulo n. Moreover, they both satisfy the associative andcommutative properties.






Z/nZ = {0, 1, 2, . . . , n− 1}, read “Z mod n Z”.









Z/nZ = {0, 1, 2, . . . , n− 1}, read “Z mod n Z”.




Modular arithmetic


addition: a + b := a + b, and

multiplication: a ∗ b := a ∗ bare well-defined binary operations on the set of congruence classesmodulo n. Moreover, they both satisfy the associative andcommutative properties.

Binary operation: a function A×A?−−→ A, written (a, b) 7→ a ? b.

Some examples: Some non-examples∗ Addition and multiplication ∗ Dot and scalar products on Rn

on Z,Q,R,C,Z>0. ∗ Subtraction on Z>0.∗ Division on C×.∗ Multiplication on Mn(R).∗ Cross products on R3.∗ Division on R>0.

Another example: Define ? on R by a ? b = ab− b2 + 5.

Modular arithmetic



multiplication: a ∗ b := a ∗ bare well-defined binary operations on the set of congruence classesmodulo n. Moreover, they both satisfy the associative andcommutative properties.Binary operation: a function A×A

?−−→ A, written (a, b) 7→ a ? b.

Some examples: Some non-examples∗ Addition and multiplication ∗ Dot and scalar products on Rn



Modular arithmetic




?−−→ A, written (a, b) 7→ a ? b.Some examples: Some non-examples∗ Addition and multiplication ∗ Dot and scalar products on Rn



Modular arithmetic




?−−→ A, written (a, b) 7→ a ? b.Some examples: Some non-examples∗ Addition and multiplication ∗ Dot and scalar products on Rn



A group is a pair (G, ?) consisting of a set G and a binaryoperation ? on G such that:

1. ? is associative: (x ? y) ? z = x ? (y ? z).

2. There is an identity element e ∈ G, which satisfies

e ? x = x = x ? e for any x ∈ G.

3. Every element of G has an inverse: for any x ∈ G, there is anelement x−1 such that x ? x−1 = e = x−1 ? x.

Examples:(R,+) “The real numbers form a group under addition”

(GLn(C), ·) “The invertible matrices form a group under multiplication”

(Z/6Z,+) “The integers modulo 6 form a group under addition”

Vocabulary:

I When checking that x ? y ∈ G (part of “well-defined”), we arechecking that “G is closed under ?.”

I If (G, ?) is also commutative, then we say (G, ?) is abelian.






Examples:(R,+)

“The real numbers form a group under addition”



Vocabulary:











Vocabulary:









(GLn(C), ·)

“The invertible matrices form a group under multiplication”


Vocabulary:











Vocabulary:










(Z/6Z,+)

“The integers modulo 6 form a group under addition”

Vocabulary:











Vocabulary:











Vocabulary:











Vocabulary:



More examples of groups

Note that, for example, Q, R, and C aren’t groups undermultiplication

(0 doesn’t have a multiplicative inverse!).Fix: Let

Q× = Q6=0 = Q− {0}, R× = R 6=0 = R− {0}

C× = C 6=0 = C− {0} (recall rationalizing denominators)

(C×, ∗) “The non-zero complex numbers form a group under multiplication”(R×, ∗) “The non-zero real numbers form a group under multiplication”(Q×, ∗) “The non-zero rational numbers form a group under multiplication”((Z/nZ)×, ∗) “The integers relatively prime to n, modulo n,

form a group under multiplication”(See homework)

What is Z×?


Note that, for example, Q, R, and C aren’t groups undermultiplication (0 doesn’t have a multiplicative inverse!).

Fix: Let

Q× = Q6=0 = Q− {0}, R× = R 6=0 = R− {0}




What is Z×?


Note that, for example, Q, R, and C aren’t groups undermultiplication (0 doesn’t have a multiplicative inverse!).Fix: Let

Q× = Q6=0 = Q− {0}, R× = R 6=0 = R− {0}




What is Z×?



Q× = Q6=0 = Q− {0}, R× = R 6=0 = R− {0}




What is Z×?



Q× = Q6=0 = Q− {0}, R× = R 6=0 = R− {0}




What is Z×?


1. Zn,Qn,Rn,Cn under addition infinite

2. Q×,R×,C× under multiplication infinite

3. Z/nZ under addition finite

4. (Z/nZ)× = {a ∈ Z/nZ | a is relatively prime to n} undermultiplication. (homework) finite

5. Mn(F ) under addition, where F = Q,R,C etc. infinite

More to come:

6. More general linear groups GLn(F )

7. Dihedral groups D2n

8. Symmetric groups Sn

9. Quaternian group Q8.







More to come:











More to come:











More to come:











More to come:











More to come:





Algebraic structures

∗ A group is a set G with a binary operation ? satisfyingassociativity, and the existence of an identity and inverses.

∗ A monoid is a set M with a binary operation ? satisfyingassociativity and the existence of an identity. (e.g. (Z,×))

∗ A semigroup is a set S with a binary operation ? satisfyingassociativity (e.g. (Z>0,+)).

∗ A field is a set F with two binary operations, + and ×, suchthat (F,+) and (F − {0},×) are abelian groups, and thedistributive property holds between + and ×. (e.g. R or Z/pZ)

∗ A vector space (over a field F ) is a an abelian group (V,+) thathas a faithful action on it by F that satisfies the distributiveproperties. (i.e. for f ∈ F , u, v ∈ V , f · (u + v) = f · u + f · v).(e.g. F = R, V = Rn)

∗ A ring (with identity) is a set R with two binary operations, +and ×, such that (R,+) is an abelian group, and (R,×) is amonoid, and the distributive property holds. (e.g. Z or Z[x])

∗ An algebra (over a field F ) is a ring A such that (A,+) forms avector space over F and the distributive properties hold. (e.g.F = R, A = R[x]) . . . and many many others.










































∗ A group is a set G with a binary operation ? satisfying associativity,and the existence of an identity and inverses.

∗ A monoid is a set M with a binary operation ? satisfying associativityand the existence of an identity. (e.g. (Z,×))


∗ A field is a set F with two binary operations, + and ×, such that(F,+) and (F − {0},×) are abelian groups, and the distributiveproperty holds between + and ×. (e.g. R or Z/pZ)

∗ A vector space (over a field F ) is a an abelian group (V,+) that has afaithful action on it by F that satisfies the distributive properties. (i.e.for f ∈ F , u, v ∈ V , f · (u+ v) = f · u+ f · v). (e.g. F = R, V = Rn)

∗ A ring (with identity) is a set R with two binary operations, + and ×,such that (R,+) is an abelian group, and (R,×) is a monoid, and thedistributive property holds. (e.g. Z or Z[x])

∗ An algebra (over a field F ) is a ring A such that (A,+) forms a vectorspace over F and the distributive properties hold. (e.g. F = R,A = R[x])

. . . and many many others.


∗ A group is a set G with a binary operation ? satisfying associativity,and the existence of an identity and inverses.

∗ A monoid is a set M with a binary operation ? satisfying associativityand the existence of an identity. (e.g. (Z,×))


∗ A field is a set F with two binary operations, + and ×, such that(F,+) and (F − {0},×) are abelian groups, and the distributiveproperty holds between + and ×. (e.g. R or Z/pZ)

∗ A vector space (over a field F ) is a an abelian group (V,+) that has afaithful action on it by F that satisfies the distributive properties. (i.e.for f ∈ F , u, v ∈ V , f · (u+ v) = f · u+ f · v). (e.g. F = R, V = Rn)

∗ A ring (with identity) is a set R with two binary operations, + and ×,such that (R,+) is an abelian group, and (R,×) is a monoid, and thedistributive property holds. (e.g. Z or Z[x])

∗ An algebra (over a field F ) is a ring A such that (A,+) forms a vectorspace over F and the distributive properties hold. (e.g. F = R,A = R[x]) . . . and many many others.

For now: Groups.

Two notational simplifications: When extra clarity is not reallyneeded,

1. we often write xy in place of x ? y, and call generic binaryoperations “products”;

2. we write G in place of (G, ?), e.g. “let G be a group”.Ambiguity! What do I call the set now??The underlying set of a group G is the set of elements in G.

For now: Groups.



2. we write G in place of (G, ?), e.g. “let G be a group”.

Ambiguity! What do I call the set now??The underlying set of a group G is the set of elements in G.

For now: Groups.



2. we write G in place of (G, ?), e.g. “let G be a group”.Ambiguity! What do I call the set now??

The underlying set of a group G is the set of elements in G.

For now: Groups.



2. we write G in place of (G, ?), e.g. “let G be a group”.Ambiguity! What do I call the set now??The underlying set of a group G is the set of elements in G.

For now: Groups.

TheoremLet G be a group.

1. There is only one identity element e in G.

Notation: when writing xy in place of x ? y, we also mightwrite 1 or 1G in place of e.

2. For a given g ∈ G, g−1 is unique.

3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).

5. For any a1, a2, . . . , an ∈ G, the value of a1a2 · · · an isindependent of how the expression is bracketed (generalizedassociative law).

6. For any x, y ∈ G, there are unique elements z, z′ ∈ G so thatxz = y and z′x = y.

For now: Groups.


1. There is only one identity element e in G.Notation: when writing xy in place of x ? y, we also mightwrite 1 or 1G in place of e.


3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



For now: Groups.




3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



For now: Groups.




3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



For now: Groups.




3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



For now: Groups.




3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



For now: Groups.




3. (a−1)−1 = a.

4. For any x, y ∈ G, (xy)−1 = (y−1x−1).



You try: Let X be a set and let ? be an associative binaryoperation on X (G is a semigroup under ?). Let e be an identityfor ? (so that e ? x = x ? e = x for all x ∈ X).

(1) Suppose you have an element z 6= e such that x? z = z ?x = zfor all x ∈ X, then z does not have a multiplicative inverse.

(2) Show that if z is as in (1), and xy = z for some y 6= z, then xdoes not have a multiplicative inverse.

(3) Write the multiplication tables for the sets Z/nZ forn = 3, 4, 5, and 6 under multiplication. Identify maximalsubsets of each example that is closed under multiplicationand, further, forms a group under multiplication.

Powers Still, let G be a group. . .

Recall that the (generalized) associative property says for anyx1, . . . xn ∈ G, the value of x1x2 . . . xn does not depend on on theexpression is parenthesized.

Define:

xn = x · x . . . x and x−n = (x−1)n.

TheoremFor any x ∈ G,

x−n = (xn)−1, xmxn = xm+n, and (xm)n = xmn,

for all integers m and n. (D&F, 1.1#19)

Order Still, let G be a group. . .

Our goal is to be able to ask and answer questions like:“What are some properties of groups?”“How are two groups similar or different? ”

DefinitionThe order of G, denoted |G|, is the size of the underlying set.

For any element x ∈ G, if xn = 1 for some n ∈ Z>0, we say theorder of x is the smallest such n.

Theorem

1. An element x ∈ G has order 1 if and only if x = 1.

2. xm = 1 if and only if |x| divides m.





Theorem







Theorem



Groups of symmetries

16

5

4 3

2

→

s

61

2

3 4

5

↓

r

65

4

3 2

1


16

5

4 3

2

→

s

61

2

3 4

5

↓

r

65

4

3 2

1


16

5

4 3

2 →

s

61

2

3 4

5

↓

r

65

4

3 2

1


16

5

4 3

2

→

s

61

2

3 4

5

↓

r

65

4

3 2

1


16

5

4 3

2

→

s

61

2

3 4

5

↓

r

65

4

3 2

1


16

5

4 3

2

→

s

61

2

3 4

5

↓

r65

4

3 2

1

1 r r2 r3 r4 r5

16

5

4 3

2

65

4

3 2

1

54

3

2 1

6

43

2

1 6

5

32

1

6 5

4

21

6

5 4

3

s rs r2s r3s r4s r5s

61

2

3 4

5

12

3

4 5

6

23

4

5 6

1

34

5

6 1

2

45

6

1 2

3

56

1

2 3

4

You try: Draw the symmetries for the triangle.(1) How many symmetries are there?(2) If we call the move “rotate clockwise” r, what is the order of r? Isthere a way to write r−1 in terms of some positive power of r?(3) If we call the move “flip across a vertical axis” s, what is the order ofs? Is there a way to write s−1 in terms of some positive power of s?(4) Note that rasb means “flip b times and then rotate a times” (readactions right to left like function composition). Now label each of thesymmetries by some rasb. Then label each of the symmetries by somesbra, and then by sbr−a, and compare all three forms.Repeat parts (1)–(4) for the square.

1 r r2 r3 r4 r5

16

5

4 3

2

65

4

3 2

1

54

3

2 1

6

43

2

1 6

5

32

1

6 5

4

21

6

5 4

3


61

2

3 4

5

12

3

4 5

6

23

4

5 6

1

34

5

6 1

2

45

6

1 2

3

56

1

2 3

4


1 r r2 r3 r4 r5

16

5

4 3

2

65

4

3 2

1

54

3

2 1

6

43

2

1 6

5

32

1

6 5

4

21

6

5 4

3


61

2

3 4

5

12

3

4 5

6

23

4

5 6

1

34

5

6 1

2

45

6

1 2

3

56

1

2 3

4


1 r r2 r3 r4 r5

16

5

4 3

2

65

4

3 2

1

54

3

2 1

6

43

2

1 6

5

32

1

6 5

4

21

6

5 4

3


61

2

3 4

5

12

3

4 5

6

23

4

5 6

1

34

5

6 1

2

45

6

1 2

3

56

1

2 3

4


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Modern Algebra 1 - City University of New York...Last time: Modular arithmetic. For any two integers...

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