課本：

https://faculty.math.illinois.edu/~clein/ab-alg-book.pdf

本筆記包含了課件里幾乎所有知識點，一共13個week

Week1

概念

• N = natural numbers
• Z = integers
• Q = rational numbers
• R = real numbers

集合運算：

• subsets: A?B
• intersections: A∩B
• unions: A∪B
• difference: A-B，如N-E = odd natural numbers
• direct product:?A×B = {(a, b)|a∈A, b∈B}

a function(map or mapping): is a rule that assigns an element of B to every element of A (Given two sets A & B). σ: A→B:

• A = domain of σ
• B = range of σ
• ?a∈A,?σ(a)∈B,?σ(a)=value of σ at a (the image of a)
• 原像 preimage of C: If C?B, = {a∈A | σ(a)∈C}
• fiber of b==?{a∈A | σ(a)=b}，given b∈B

composition: τ°σ?: A→C, ?a∈A, given σ: A→B,?τ: B→C，即τ°σ(a)=τ(σ(a))

A function σ: A→B is called

• 單射 injective(one-to-one):??a1, a2∈A, if σ(a1) = σ(a2) then a1 = a2
• 滿射 surjective(onto):??b∈B, ?a∈A s.t. σ(a) = b
• 雙射 bijective: if it’s surjective & injective

identity function on A: : A→A is defined by ?for all a∈A

inverse?:
A function σ: A→B is a bijection if ? a function τ: B→A s.t.

• σ°τ = idB
• τ°σ = idA

Such τ is unique, called inverse of σ,??= τ

集合的基數 Cardinalities of Sets |A|=#of elements, 如empty set|?| = 0

Sets of Functions?= {σ : A→B |?σ a function}, A, B are sets, 如C(R,R)={continuous functions σ: R→R}???

An operation on a set A is a function *:?, 如+, · ,- on Q or R,?

associativity:
* is associative if (a * b) *?c = a *?(b *?c), ?a, b, c∈A. °、+ is associative; - is not associative

Given a set A, a relation on A is a subset R ? A×A. Instead of (a, b)∈R, we write a ~ b

~ a relation on A is?an equivalence relation if it satisfies all three:

• reflexive if a ~ a ?a∈A
• symmetric if a ~ b ? b ~ a ?a, b ∈ A
• transitive if a ~ b and b ~ c, then a ~ c ?a, b, c ∈ A

~f:?f: X → Y is a function, define ~f on X by a ~f b if f(a) = f(b)

partition:?X a set, a partition of X is a collection Ω?of subsets of X s.t.

?A, B∈Ω either A = B or A ∩ B = ?

the equivalence class?of x [x] = {y∈X | y~x},?X is a set with?an equivalence relation ~,?x∈X
Equivalence classes are fibers

對稱群 the symmetric group of X / the permutation group of X?Sym(X) = {f: X→X | f is a bijection}???,?X?nonempty set, 即Sym(X) are bijections from X to itself.
When X = {1, ..., n}, n∈Z, write = Sym(X) symmetric group on n elements, 如S3 = Sym({1, 2, 3}),?有n!個元素
https://zh.wikipedia.org/wiki/%E7%BD%AE%E6%8F%9B

Cycle Decompositions of Permutations:

disjoint cycles:
σ = τ1°τ2 = τ2°τ1, τ1&τ2 are disjoint cycles, disjoint cycles commute
τ1、τ2 in cycle notation:

• τ1 = (1 2 5) = (2 5 1) = (5 1 2)
• τ2 = (3 4) = (4 3)

disjoint cycle notation for σ:?σ = (1 2 5) ° (3 4) = (1 2 5)(3 4)

σ∈Sym(X):

• σ^2=σ°σ, σ^3=σ°σ°σ
• , r>0
• r,s∈Z,?

性質

Suppose : σ: A→B,?τ: B→C are functions,

• If σ,?τ?are injective, then τ°σ is injective;
• If σ,?τ are surjective, then τ°σ is surjective;
• If σ,?τ are bijections, then so is τ°σ.

Equivalence relations are the same as partitions:
X any nonempty set, ~ an equivalence relation on X. Then the set of equivalence classes X/~?= {[x]|x∈X} is a partition.
Conversely, given a partition Ω of X, there exists a unique equivalence relation ~ s.t. X/~?= Ω

If ~ is an equivalence relation on X, define?π: X→X/~ by π(x) = [x], then ~π = ~

For any nonempty set X, ° is an operation on Sym(X), and

• ° is associative
• ∈Sym(X), ?° σ?= σ ° = σ
• ?σ∈Sym(X), ∈Sym(X),?σ is a 置換?permutation,?X = given marvels,?”permutation” of marvels is a rearrangement

Every permutation is a composition of disjoint?cycles, uniquely

Week2

概念

b|a:?b divides a,?if ?m∈Z, so that a = bm, b|a,?Suppose a,b∈Z, b≠0

b?a:?b does not divide a

prime:?p > 1, p ∈ Z is called prime if the only divisors are ±1, ±p

Prime Factorization of Integers:?Any integer a > 1 has a prime factorization:
?where p_i > 1 is prime,?k_i∈Z+, ? i = 1, ..., n, p_i ≠ p_j ?i≠j

GCD:
Given a,b∈Z,?a≠0, b≠0, the greatest common divisor of a and b is c∈Z, c>0 s.t.

• c|a and c|b
• if d|a and d|b, then d|c

If c exists, it’s unique.?Write c = gcd(a, b).

歐幾里得算法?Euclidean Algorithm: Given a,b∈Z, b≠0, then ?q,r∈Z s.t. a = bq + r,?0 ≤ r < |b|

互質 relatively prime:?gcd(b, c)=1

模n同余 congruence modulo n:?a relation on Z by a ≡ b mod n?if n|(a-b).
Congruence modulo n is an equivalence relation??n∈Z+

同余類?congruence class of a mod n?=[a]=equivalence of a modulo n = {b∈Z | b ≡ a mod n}
[a] = {a + kn | k∈Z}

={?| a∈Z}, 即integers mod n
?a∈,?there are exactly n congruence classes modulo n:?= {[0], [1], [2], ..., [n-1]}

Fix n∈Z. Define + & · on

• [a] + [b] = [a + b]
• [a][b] = [ab]

Addition and Multiplication Tables:

Say [a] ∈ is a unit or is invertible if ? [b] ?∈ ? so that [a][b] = [1].
inverse of [a]??= [b] is unique

n≥2, = {[a] ?∈ ?| [a] is a unit}
?n≥2,??= {[a] ?∈ ?| gcd(a, n)=1}
If p≥2 is prime,? = {[1],...,[p-1]},?=p-1

Euler phi-function φ(n)=

Complex Numbers:

• addition & multiplication?are commutative and associative
• 復共軛性?Complex conjugation: z = a + bi, zˉ = a - bi
• Absolute value: |z| = √(a^2 + b^2)
• 復平面?complex plane
• -z = -a -?bi is the additive inverse of z =?a + bi
• ? z ∈ C\{0}, the multiplicative inverse is?

性質

Basic Properties of Integers:

• addition & multiplication are associative and commutative operations
• 0∈Z is the additive identity: 0 + a = a ?a∈Z
• ?a∈Z,?the additive inverse -a = (-1)a : a + (-1)a = a-a = 0
• 1∈Z?is the multiplicative identity: 1a = a??a∈Z
• multiplication distributes over addition: a(b + c) = ab + ac??a,b,c∈Z
• N = {0, 1, 2, ...}, Z+ = {1, 2, ...} are closed under addition & multiplication
• ?a,b∈Z,?a≠0, b≠0, |ab| ≥ max{|a|, |b|}, strict inequality iff |a| > 1 & |b| > 1

?a,b∈Z

• if a≠0, then a|0
• if a|1, then a = ±1
• if a|b & b|a, then a = ±b
• if a|b & b|c, then a|c
• if a|b & a|c, then a|(mc + nb) ?m,n∈Z

Existence of Greatest Common Divisor:
?a,b∈Z not both 0,?gcd(a, b) exists and is the smallest positive integer in the set M = {ma + nb | m,n∈Z}.
In particular, ?m0,n0∈Z s.t. gcd(a, b) = m0a + n0b.

Divisibility:

• Suppose a, b, c ∈ Z. If?gcd(b, c)=1 and b|ac, then b|a
• a, b, c ∈ Z, p > 1 prime. If p|ab, then p|a or p|b

Addition and Multiplication on Congruence Classes:
a,b,c,d,n∈Z,?n > 1, a ≡ c mod n, b ≡ d mod n, then

• a + b ≡ c + d mod n
• ab ≡ cd mod n

Properties of the Addition and Multiplication on Congruence Classes:
a,b,c,n∈Z,?n ≥?1, then in?

• addition & multiplication are commutative & associative operations
• [a] + [0] = [a]
• [-a] + [a] = [0]
• [1][a]=[a]
• [a]([b]+[c]) = [a][b]+[a][c]

Chinese Remainder Theorem:

If m, n, k > 0, n = mk, gcd(m, k) = 1, then

• by?is a bijection
• Then
• Then φ(n)=φ(m)φ(k)

If n ∈ Z+, n =prime factorization. Then

Week3

概念

復數?Complex Numbers:

• Re(z) real part、Im(z) imaginary part
• θ?= angle (or argument of z)
• addition: geometrically is parallelogram law
• multiplication:
  • z = r(cosθ?+ i sinθ) =?
  • w = s(cosψ?+ i sinψ)
  • zw =?|z||w|(cos(θ+ψ) + i sin(θ+ψ))

域 field: is a nonempty set F with?2 operations, addition written a + b (for a, b ∈ F) and multiplication written a · b = ab (for a, b ∈ F) s.t.

• addition and multiplication are associative and commutative
• Distributive: a(b + c) = ab + ac, ?a,b,c∈F
• ?an additive identity: 0 ∈ F s.t. 0 + a = a, ?a∈F
• ?a∈F, there is an additive inverse -a s.t. a + (-a)=0, ?a∈F
• ?a multiplicative identity: 1 ∈ F s.t. 1a = a, ?a∈F, 1 ≠ 0
• ?a∈F, a ≠ 0, a has a multiplicative inverse?

子域?subfield: Call such K a subfield of F: suppose F is a field and K ? F s.t.

• 0, 1 ∈ K
• ?a,b∈K, a + b, ab, -a ∈ K;?if a ≠ 0, a^-1 ∈?K,
  i.e. K is closed under addition, multiplication, additive inverse, and multiplicative inverse, respectively

多項式?polynomia: A polynomial over F in variable x is a formal sum: a0 + a1x + a2x^2 + ... + a_n x^n =,?where n≥0 is an integer, a0, a1, ..., a_n ∈ F, (x^0=1, x^1=x).
polynomial is a sequence eventually 0, i.e. ?n≥0 s.t. ?m>n, a_m = 0
Constant polynomials:?f = a

F[x]={|n≥0, n∈Z, a0,...,a_n∈F} = polynomial ring over the field F
F∈F[x] is the constant polynomials.?0∈F add. identity,?1∈F mult. identity

性質

Fundamental Theorem of Algebra:

• Suppose f (x) = a0 + a1x + ... + a_n x^n is a nonconstant polynomial, coecients a0, a1, ..., a_n ∈ C. Then f has a root in C.
• Every nonconstant polynomial w/ coecients a0, a1, ..., a_n(a_n≠0) ∈ C can be factored as f(x) ==a_n(x-a1)(x-a2)···(x-a_n).
• If f (x) = a0 + a1x + ... + a_n x^n is a nonconstant polynomial, coecients a0, a1, ..., a_n ∈ R,?a_n≠0.?Then f can be expressed?as a product of linear and 二次多項式?quadratic polynomials.

F a field, a, b ∈ F, then

• If a + b = b then a = 0
• If ab = b and b ≠?0, then a = 1
• 0a = 0
• If a + b = 0, then b = -a
• If a ≠ 0 and ab = 1, then b = a^-1

Suppose K ? F is a subfield of a field F, then the operations of F make K into a field

Suppose F is any field. Then

• Addition and multiplication are commutative & associative operations on F[x]
• Multiplication distributes over addition:?f(g + h) = fg + gh for all f, g, h ∈ F[x]
• 0 ∈?F?is additive identity in F[x]: ?f ∈ F[x], f+0=0
• ?f ∈ F[x], -f = (-1)f is the additive inverse: f + (-1)f = 0
• 1 ∈?F, is the multiplicative identity in F[x]: 1f = f ?f∈F[x]

Every nonzero constant polynomial has a mult. inverse

Week4

概念

deg(f) = degree of f=

• 0, if f is constant, f≠0
• n,?if a_n ≠?0 in above (a_n = leading coefficient)
• -∞, if f = 0

-∞ + a = a + (-∞) = -∞ ?a∈Z∪{-∞}

Irreducible Polynomials:
if f = uv, u, v ∈ F[x], then either u or v is a unit (i.e., constant ≠ 0).
f ∈ F[x], nonconstant is irreducible, whenever we write f = uv, then either u or v is constant.
Last time, f is monic nonconstant, then f = p1 ··· p_k , where p1, p2, ..., p_k are irreducible.

Divisibility: f , g ∈ F[x], f≠0, f divides g, f|g means ?u∈F[x] s.t. g = fu

GCD gcd: Given polys f , g ∈ F[x], nonzero a gcd of f, g is a polynomial h ∈ F[x] s.t.

• h|f and h|g and
• If k|f and k|g, then k|h

根 Roots: Define?a function f : F → F, f∈F[x], f=, ?a∈F, f (a) =. Say a ∈ F is a root of f if f(a)=0

重數 Multiplicity of Roots: If α is a root of f , say its multiplicity is m if x-α?appears m times in irreducible factorization

向量空間?Vector Space: A vector space over a field F is a set V w/ an operation + : V × V → V and function F × V → V called scalar multiplication:

• Addition is associative & commutative
• ?0∈V, additive identity: 0 + v = v ?v∈V
• 1v = v ?v∈V(where 1 ∈ F is multi. id. in F)
• ?α,β∈F, v∈V, α(βv)=(αβ)v
• ?v∈V, (-1)v = v we have v + (-v)=0
• ?α∈F, v, u ∈ V, α(v + u) = αv + αu
• ?α,β∈F, v∈V, (α+β)v = αv+βv

線性變換?linear transformation: Given two vector spaces V and W over F, a linear transformation is a function T : V→W s.t. for all a∈F and v,w∈V , we have T(av) = aT(v) and T(v+w) = T(v) + T(w)

性質

For any field F and f , g ∈?F[x], we have

• deg(fg) = deg(f) + deg(g),??,?
• deg(f+g) ≤ max{deg(f), deg(g)}

Units in Polynomial Ring: Let F be a field, f ∈ F. Then f is a unit(i.e., invertible) iff deg(f) = 0

existence and uniqueness of factorization into irreducibles:
Suppose F is a field and f ∈ F[x] is any nonconstant polynomial, then

• f = a p1 p2 · · · p_k where a∈F, p1, ..., p_n ∈ F[x] are irreducible 首一的?monic polynomials (monic = i.e. leading coeff. 1).
• If f = a q1 q2 · · · q_l, q_i monic irreducible, then?k=l and after reindexing, p_i = q_i ?i
• Suppose F is a field and f ∈ F[x] is nonconstant monic polynomial then f = p1 · · · p_k, where each p_i is monic irreducible.

f , h, g ∈ F[x], then

• If f ≠ 0, f|0
• If f|1, f is nonzero constant
• If f|g and g|f , then f = cg where c ∈ F×
• If f|g and g|h, then f|h
• If f|g and f|h, then ?u, v ∈ F[x], f | (ug + vh)

gcd’s are unique up to units, so the monic gcd is the gcd.

Euclidean Algorithm: Given f , g ∈ F[x], g ≠ 0, then ?q,r ∈ F[x] s.t.

• deg(r) < deg(g)
• and f = qg + r

Any 2 nonzero polynomials f , g ∈ F[x] have a gcd in F[x].
In fact among all polynomials in the set M = {uf + vg | u, v ∈ F[x]}, any nonconstant of minimal degree are gcds.

Divisibility of Irreducible Polynomials:

• If f , g, h ∈ F[x], gcd(f,g) = 1, and f|gh, then f|h
• If f ∈ F[x] is irreducible, and f|gh, then f|g or f|h

?f∈F[x] and α∈F,?there exists a polynomial q∈F[x] s.t. f = (x-α)q + f(α).
In particular, if α is a root, then (x-α)|fGiven a nonconstant polynomial f ∈ F[x], the # of roots counted w/ multiplicity is at most deg(f).

Matrix: If v1,...,v_n is a basis for a vector space V and w1,... w_m is a basis for a vector space W (both over F), then any linear transformation T : V→W determines (and is determined by) the m × n matrix,?where the entries are defined by

Week5

概念

Euclidean Geometry——等距算子?Isometries:?Isom(R^n) = {Φ: R^n →?R^n | |Φ(x)-Φ(y)| = |x-y|}, Φ a bijection

L(V, V): Suppose V is a vector?space over F, a field L(V, V) = {s:?V→V |?s linear}:

• pointwise addition + makes L(V, V) into an abelian gp
• composition is "multiplication"?– associative
• is distributive

n階方陣?: is?also a ring

GL(V) ={T∈L(V,V) | T is a bijection} = L(V,?V) ∩ Sym(V)
GL(n,?F) = {A ∈ | det(A) ≠ 0}

正交群?Orthogonal Group O(n)?={A ∈ GL(n,?F) | =I}

Group:
A group is a nonempty set G w/ an operation *: G×G→G s.t. the following holds

• * is associative
• ?e∈G an identity element e*g = g*e = g ?g∈G
• ?g∈G, ?g^-1∈G s.t. g * g^-1 = g^-1 * g = e

In a group (G, *), g, h ∈ G, write gh for g*h.
In particular, G×H, g, g'?∈ G, h, h' ∈ H, (g, h)(g', h')=(gg', hh')
e ∈ G, identity in G;?e ∈ H, identity in H

交換群/阿貝爾群 Abelian group:?* is commutative

二面體群?Dihedral Groups?Dn = {Φ ∈?Isom(R^2) | Φ(P_n) = P_n}

Subgroup?H < G:
If (G, *) is a group, then H?G is a subgroup of (G, *) if

• H≠?, and
• ? h, h'?∈ H, h*h'?∈ H and h^-1?∈ H?(∴ ? h ∈ H & h^-1?∈ H ? h * h^-1 = e ∈?H)

Ring:
A ring is a nonempty set R together with?2 operations, denoted +, ? (addition & multiplication) (R,+, ?) satisfying:

• (R,+) is an abelian group: the pointwise addition?is commutative、identity is 0、the inverse of a ∈ R is ?a
• ? is associative, (即(R,·) is a subgroup)
• distributivity: ?a, b, c ∈ R, a ? (b + c) = a ? b + a ? c & (b + c) ? a = b ? a + c ? a

(Z, +, ?) forms a ring:

• Write Z is a ring (in general if operations are "obvious ones")
• In any ring, write ab = a ? b.

commutative ring:?? is commutative
a ring with?1: there is an elt. 1 ∈ R?{0} s.t. 1?a = a?1 = a,?a∈R
A field is a commutative ring R with?1 s.t. every nonzero elt. is invertible. ?a∈R, a≠0, ?a^-1∈R s.t. a^{-1}?a = 1.

Subring:
A subring S of a ring R is a nonempty subset S ? R s.t.

• ?a, b ∈ S, a + b, ab ∈ S
• ?a ∈ S,?a ∈ S

Direct Product ?:
(h, k) ? (h', k')=(h * h', k *?k'), hh'∈G, kk'∈H

性質

Φ, ψ ∈?Isom(R^n), then Φ°ψ,?Φ^-1 ∈ Isom(R^n)
Isom(R^n) < Sym(R^n)

Every Isometry is the Composition of an Orthogonal Transformation and a Translation:
Isom(R^n) =?{?| A∈O(n), v∈R^n}

If (G, *) is a group, H?G is a subgroup, then (H, *) is a group.
If S ? R is a subring, then +, ? make S into a ring.

Properties of Group Operation: Inverse and Identity:
(G, *) be a group, g, h∈G, then (都很容易證明)

• If either g*h = h or h*g = h, then g = e
• If g*h = e,?then g = h^-1 and h = g^-1
• e^-1 = e
• (g^-1)^-1 = g
• (g*h)^-1 = h^-1 * g^-1
• g, h, k ∈ G. If g*h = k*h, then g = k;?Likewise if h*g = h*k, then g = k.
• the equations g * x = h and x * g = h have unique solutions x∈G.

Let (G, *) be a group. g∈G, n∈Z. Then defining g^0 = e, g^n = g^(n-1) * g ?n>0, g^n = (g^-1)^-n, n<0, we have

• g^n *?g^m = g^(n+m)
• (g^n)^m = g^(nm) ?n,m∈Z

For any groups (G, *) and (H, *), (G × H,??) is a group. The identity is (e_G, e_H), (g, h)^-1 = (g^-1, h^-1) ?g∈G, h∈H

Week6

概念

?(A) = all subgroups of G containing A.?let G be a group,?A ? G, A ≠ ? subset.
<A> = = = subgroup generated by A

循環群?cyclic group <a> = < {a} > =?= cyclic subgroup generated by a, a∈G,?A = {a}, a is called a generator for <a>.
Let G be a group, g∈G. Then <g>?= {g^n | n∈Z}

Order:

• G a group, |G| = order of G
• If g∈G, define |g| = |<g>| = order of g

Holomorph of a group - Groupprops

Lattice of subgroups of a group G: all subgroups with?containment data
如Z_4、Z_6和Z_12：

性質

Let G be a group, let ? be a nonempty collection of subgroups of G, then K=???????is a subgroup.

G is cyclic ??G is abelian

Let G be a group for g∈G, the following are equivalent:

• |g| < ∞
• ?n≠m in Z so that g^n = g^m
• ?n∈Z so that g^n = e
• ?n∈Z+ so that g^n = e
  ∵?If |g| < ∞, then |g| = smallest n∈Z+ so that g^n = e, and <g>?= {e, g, g^2 , ..., g^(n-1)} = {g^n|n = 0, ..., n-1}

If H < Z, then H = {0}, or H = <d>, where d = min{H∩Z+}.
Consequently, a → <a>?defines a bijection from N = {0, 1, 2, ...} → {subgroups of Z}.
Furthermore, for a, b ∈ Z+,?<a> < <b>?iff b|a.

?n≥1, if H < Z_n is a subgroup, then ? a positive divisor d of n so that H = <[d]>.
If d, d' > 0?are two divisors of n, then <[d]>?< <[d']> iff d'|d.

the order of any subgroup of?G divides the order of G

A group?G is cyclic iff it has exactly one subgroup of order?d for every divisor?d?of?|G|.

Week7

概念

AB = {ab | a∈A, b∈B}, suppose G is a group, A, B ? G nonempty subsets

Center: G a group, define center of G Z(G) = {g∈G | gh = hg ?h∈G} < G
Centralizer: For any h∈G, define the centralizer of h in G = {g∈G | gh = hg} < G
Z(G) =

群同態?Group Homomorphism: Let G, H be groups. A map Φ: G→H is called a (group) homomorphism if ?g,h∈G, Φ(gh) = Φ(g)Φ(h)

Kernel: Suppose Φ: G→H is a homomorphism the kernel of Φ is ker(Φ) = {g∈G | Φ(g) = e∈H} =?Φ^-1(e)?< G

正規子群?Normal Subgroup H?G: A subgroup H < G is normal in G if ?g∈G, gHg^-1 = H, 即gH=Hg.

群同構?Group Isomorphism?G≌H: A (group) isomorphism from G to H is a bijective homomorphism Φ: G→H. In this case, say G & H are isomorphic

自同構群 Group Automorphsim: If Φ: G→G is an isomorphism, we call it an automorphism.
For any group G, the set of all automorphisms is a subgroup Aut(G) < Sym(G)

the conjugate of H by g: If H < G is a subgroup and g∈G, we call gHg^-1 the conjugate of H by g. A conjugate of H is again a subgroup of G

conjugation by g: If G is a group, g∈G, conjugation by g is the function : G→G, given by .
?∈ Aut(G),?

: G→G, ,?G be a group, g∈G

性質

Suppose G and H are groups,?Φ: G→H a group homomorphism,?then

• Φ(e) = e
• ?g∈G, n∈Z, Φ(g^n) = Φ(g)^n
• ker(Φ) < G
• ker(Φ) ? G. Let N = ker(Φ), then ?g ∈ G, gN = Ng = Φ^?1 (Φ(g))
• Φ is injective iff ker(Φ) = {e}
• ? subgroups K < G and J < H, we have Φ(K) < H and Φ^-1(J) < G are subgroups
• Suppose Φ: G→H and ψ: H→K are homomorphisms. Then Φ°ψ: G→K is a homomorphism

Let G be a group, N < G. Then N?G iff ?g∈G, gNg^-1 ? N

Every subgroup of an abelian group is normal

Suppose Φ: G→H is an isomorphism. Then Φ^-1: H→G is an isomorphism.
If ψ: H→K is an isomorphism, then ψ°Φ: G→K is an isomorphism

Suppose G is a cyclic group, g∈G a generator

• If |G| = |g| = ∞, then Φ: Z→G given by Φ(n) = g^n is an isomorphism
• If |G| = |g| = n < ∞, then Φ([a]) = g^a, and this well defines an isomorphism Φ: Z_n→G

Cayley’s Theorem: For any group G, any g∈G, L_g: G→G is a bijection and L_?: G → Sym(G),?g ? L_g, is an injective homomorphism

For any group G, G is isomorphic to a subgroup of Sym(G)

For any integer n≥1, S_n = Sym({1, 2, ..., n}) contains an isomorphic copy of every finite group G with?|G| ≤ n

X, Y two sets, Φ: X→Y bijection, then C_Φ: Sym(X) →?Sym(Y), C_Φ (σ) = ΦσΦ^-1 is an isomorphism.

Week8

概念

魔方?Rubik’s Cube: Solving a Puzzle Using Group Theory

Sign Homomorphism on the Permutation Group: Define a homomorphism on S_n called the sign homomorphism (or parity): ε: S_n → {±1} = C_2 < C×

Definition of Sign Homomorphism?ε:
Let σ ∈?S_n, q(x1, x2, ..., x_n), σ · q(x1, x2, ..., x_n) = q(x_σ (1), x_σ (2), ..., x_σ (n))
στ?· q =?σ · (τ · q)
Define ε(σ) ∈ {±1}, s.t.?σ·p(x1, x2, ..., x_n) =?ε(σ)p(x1, x2, ..., x_n)

二面體群?Dihedral Group?D_n ? symmetries of regular n-gon in R^2,?n≥3

• D_n <?Isom(R^2)
• D_n < O(2)
• |D_n| = 2n

抽代雜談(13): 二面體群 - 知乎

Cosets of a Subgroup: let G be a group and H < G any subgroup. A left coset of H (by g) is a set the form gH = {gh | h∈ H} ? G for some g ∈ G.
https://zh.wikipedia.org/wiki/%E9%99%AA%E9%9B%86

Index of a Subgroup: the index of H in G is the number of left cosets?[G:H] = |G/H|, G/H = {gH | g∈G} = the set of left cosets.
Observe that ?g ∈ G, g = ge ∈ gH so every element of G is in some left coset.
If H < G is a subgroup of G, then G/H is a partition of G

Coset Representatives: Suppose H < G, a subset H ? G is a set of left coset representatives of H in G if ?g ∈ G, ? a unique g' ∈ H so gH = g'H

性質

Every permutation is a composition of 2-cycles
If σ ∈?S_n is a composition of k 2-cycles, ε(σ)=
If σ ∈?S_n is a k-cycle, then ε(σ)=. ∵Every k-cycle is a composition of (k-1) 2-cycles

Let G be a group, g1, g2 ∈ G, H < G. Then the following are equivalent:

• g1, g2 belong to a common coset
• g1H = g2H
• ∈ H

Lagrange’s Theorem: Let H < G be a subgroup of a finite group G. Then |G| = [G:H] |H|. Consequently

• |H| | |G|
• Let G be a finite group. g ∈ G, |g| | |G|
• 可用于證明Fermat's Little Theorem
• Let H < K < G be subgroups of a finite group G, Then [G:H] = [G:K][K:H]

Let p > 1 be any prime, G a group of order |G| = p. Then G≌Z_p, i.e. G is cyclic. And??g∈G, g≠e is a generator.

Let G be a group and N < G a subgroup. Then the following are equivalent:

• N?G
• ?g ∈ G, gN = Ng
• ?g ∈ G, ?h ∈ G so that gN = Nh

Week9

概念

商群 Quotient Group、factor group?G/N = ”G mod N” = quotient group of G by N.
quotient homomorphism?π:?G →?G/N, π(g) = gN

環同態?ring homomorphism: 比群的同態要求更高

性質

(group G is abelian ??G/H is abelian)

If G is a group, N?G, g,h∈G, then

• (gN)(hN) = ghN.
• product of cosets makes G/N into a group.
• Furthermore π : G → G/N, π(g) = gN is a surjective homomorphism, with?ker(π) = N.

1st Isomorphism Theorem: Suppose Φ: G→H is a homomorphism, N = ker(Φ). Then there exists a unique isomorphism : G/N →?Φ(G) < H s.t.?π=Φ,?π: G → G/N quotient homomorphism ? G/N ≌ Φ(G)

R/Z ≌ S^1

Suppose H,N<G, N?G, then HN = NH

Let H,N<G,?then HN < G iff HN = NH, ∴易證The product of a subgroup and a normal subgroup is a subgroup

2nd isomorphism theorem/ Diamond isomorphism theorem: Suppose H,N < G, N?G. Then N?HN, H∩N?H, HN/N ≌ H/H∩N.

Let S(H) = {L < H|L is a subgroup},?S(G, N) = {J < G|J subgroup N < J}. Suppose Φ: G→H is a surjective?homomorphism,?N = ker(Φ), then Φ_*(J) = Φ(J) defines a map Φ_*: S(G, N) → S(H),?Φ^*(L) = Φ^-1(L) defines a map Φ^*: S(H) → S(G, N) which are inverse bijections.
These bijections preserve normality: K<H is normal iff?Φ^-1(K)<G is normal.

Suppose Φ: G→H is a surjective homomorphism, K?G,?ker(Φ) < K (Note K?G,?Φ(K)?H),?G/K ≌ H/Φ(K)

G is a group, N,K < G, NK < G. If N,K?NK and N∩K = {e}. Then Φ: N×K →?NK,?Φ(n, k) = nk is an isomorphism.
G is a group, N,K ? G s.t. NK = G, N∩K = {e}. Then Φ: N×K →?G,?Φ(n, k) = nk, is an isomorphism.

N1, N2, ..., N_k ??G, , ?i = 1, ..., k,? = {e}, then G ≌ N1×N2×···×N_k.
N1, N2, ..., N_k ??G, ,??i = 1, ..., k, = {e}, then Φ: N1×N2×···×N_k →?G is an isomorphism.

Week10

概念

半直積 Semidirect Product given by Conjugation:
Suppose G is a group, N, H < G, and N?G and N∩H = {e} and NH = G,?Let Φ: N×H → NH = G,?Φ(n, h) = nh, not a homomorphism in general, Φ is a bijection
Suppose N, H are groups and α: H →?Aut(N) () is a homomorphism

(n, h)(n', h')=(n α_h(n'), hh'), (n, h),(n', h') ∈ N×H,?N×H with this operation is called semidirect product of N & H by α N ?_α H

With the set up above, N ?_α H is a group. and N0 = N×{e}, H0 = {e}×H are subgroups N0≌N, H0≌H, N0?N ?_α H, N0∩H0 = {e}, N0H0 = N ?_α H

Group Action: Let G be a group, X ≠ ? a set. Then an action of G on X is a function G × X → X, (g, x) ? g ? x satisfying 2 properties

• e ? x = x,??x ∈ X
• g ? (h ? x) = gh ? x, ?g, h ∈ G, x ∈ X

等價定義：?g ∈ G, α_g:?X → X, α_g(x) = g ? x,?α:?G → Sym(X) is a homomorphism

Orbit: G × X → X is an action. orbit of x ∈ X is O_G (X) =?G ? x = {g ? x∣g ∈ G}
Orbits Form a Partition

action is transitive if ?x, y ∈ X, ?g∈G s.t. g ? x = y

If x ∈ X, stab_G (x) = stabilizer of x in G = {g ∈ G∣g ? x = x}

kernel of action = {g ∈ G∣g ? x = x, ?x ∈ X}.

性質

Suppose G is a group,?H,K < G, H ? G,?H ∩ K = {e}. Then the map?φ:?H ?c K?→ HK < G is an isomorphism, φ(h, k) = hk,?c: K → Aut(H),?c_k (h)=khk^?1

Suppose G is a group, X ≠ ? a set. If G × X → X is an action, then ?g ∈ G, α_g:?X → X given by α_g (x) = g ? x, is a bijection and defines a homomorphism α: G → Sym(X).
Conversely, if α: G→Sym(X) is a homomorphism, then g ? x = α_g (x) defines an action G×X → X

Week11

概念

Platonic solids:

性質

Orbit-stabilizer Theorem:?Suppose G acts on X. Let x ∈ X, H = stab_G (x). Then there is a?bijection θ:?G/H → G?x,?given by: aH ? a ? x & θ(g ? aH) = g ? θ(aH)

If G × X → X is an action, x ∈ X, then |G ? x|?= |G/stab_G (x)|?= |G| / |stab_G (x)|

Class Equation: |G|?=?

G3 ? S4

G4 ? S4 × {±1}
< G4, ?= {Φ∈G4 |?Φ?preserves orientation} = ker(det)?G4

Ru ? ( × ) ? ((A8 × A12) ? Z2)
Ru ? ker(Φ_C × Φ_E) ? ((A8 × A12) ? Z2)

There are configurations of a Rubik’s cube that have never been realized, anywhere ever.

Week12

概念

p-group:?A group G is a p-group (p a prime) if ?g ∈ G, |g|?= p^k some k ≥ 0
Order of Finite p-group:?If G is a p-group, p prime & |G|?< ∞, then |G|?= p^k some k ≥ 1

Zero Divisor:?If R is a ring, an element a ∈ R is called a zero divisor if a ≠ 0, and ?b ∈ R, b ≠ 0 s.t. ab = 0.

整環/整域?Integral Domain:?A commutative ring R w/ 1(≠ 0) is called an integral domain if R has no zero divisor.

Q?(R) = {(a,b)∣a,b∈R, b≠0}, suppose R is an integral domain.
Define ～ on Q?(R) by (a,b) ～ (a',b') ? ab'?= a'b.?This is an equivalence relation.
Q(R) = equivalences of Q?(R) w.r.t. ～, and the equivalence class of (a, b) denoted .
Q(R) = field of fraction of R

商域?Quotient Field: Suppose R comm. ring w/ 1,?An ideal in R is a subring I ? R s.t. ?a ∈ I, r ∈ R, ar ∈ I. π:?R → R/I is a ring, and π is a surjective ring homomorphism. (a + I)(b + I) = ab + I

Maximal Ideal:?Say that I ? R is maximal if I ≠ R, and if I ? J ? R, then J = I or J = R

主理想?Principal Ideal:?R comm. ring w/ 1, a∈R,?the principal ideal generated by a is ((a)) = smallest ideal containing a.

性質

Suppose G is a group with?|G|?= p a prime then G ? Z_p

Cauchy’s Theorem:?If G is a group, p is prime, p | |G|, then ?g∈G s.t. |g|?= p.

Suppose p is a prime, G a group of order p^k, some k ≥ 1. Then for any action G × X → X on a finite set X, ∣Fix_X (G)∣ ≡ ∣X∣ mod p, Fix_X (G) = {x∈X |?g?x = x ?g ∈ G} ? X

Sylow Theorems: G a finite group, |G|?= p^k m, p prime, p ? m
1st: ? 0 < i ≤ k, ? subgroup H < G w/ |H|?= p^i
2nd: Let P < G be a Sylow p-subgroup (i.e. |P|?= p^k ) and H < G any p-subgroup (i.e. |H|?= p^i some i). Then, ?g∈G, s.t. gHg^?1 < P. Any 2 Sylow p-subgroups are conjugate. ((1 2) not conj. to (1 2)(3 4))
3rd: The # of Sylow p-subgroups? satisfies 2 conditions:?≡ 1 mod p,? | |G|/p^k

If G is a finite group, P < G a Sylow p-subgroup. Then = 1 iff P ? G.

Suppose p > q ≥ 2 are distinct prime, G is a group of order |G|?= pq.

• If q ? (p ? 1), then G ?
• q∣p ? 1: then G is isomorphic to one of 2 groups

For any prime p ≥ 2,

If F is a field and R ? F is a subring, then R is an integral domain.

If R is an integral domain, then the following defines operations on Q(R):


With these operations, Q(R) is a field and the function r → r /1 defines injective homomorphism.

If R is a comm. ring w/ 1 and I ? R is an ideal, then R/I is a field iff?I is maximal

If F is a field, then it has no nonzero ideals, i.e., I ? F any ideal then I = F or I = {0}.

Suppose R comm. ring w/ 1, a ∈ R. Then ((a)) = {ra | r∈R}.

If F is a field, R any ring, then a hom Φ:?F → R is either surjective or identically 0.

If F & K are fields, Φ: F → K is a nonconstant hom., then restricts to a group hom. Φ | F^x: F^x → K^x . Therefore Φ(1) = 1 (so Φ is unital).

Week13

概念

Finite Extension Field:?F,K fields, F(subfield) ? K(extension), K a vector space over F, if = [K:F] < ∞, K is a finite extension.

Roots in an Extension Field: f ∈?F[x] ? K[x], f has no roots in F (i.e., no elt. α∈F, f(α) = 0) but can happen that f has a root in K.

Subfield Defined by Adjoining an Element:?Suppose F ? L extension of F & β∈L. Write F(β) ? L smallest subfield of L containing F & β.

Galois Group:?Suppose F ? K is an extension. The Galois group of K over F is the group Aut(K,F) = {σ:?K→K |?σ automorphism σ(a) = a ?a∈F}

性質

If p ∈ F[x] is irreducibe polynomial, then F[x]/((p)) is a field (i.e.?((p)) is maximal)

π:?F → F/((p)) = K, α?= π(x) ∈ K
Suppose F,p,K,α are as above. Then α is a root of p. Moreover [K:F] = n?= deg(p) and every elt of K can be expressed uniquely as , c_0, ..., c_{n?1} ∈ F.

Suppose F ? L, p ∈ F[x] irreducible, s.t. p has a root β in L. So, p has a root in F(β). If K is the extension from Theorem 5.1.1 and α?∈ K is the root of p. Then K ? F(β) by isomorphism a_0 + a_1?α?+ a_2 α^2 + ... + a_{n?1} α^{n?1} ? a_0 + a_1?β?+ a_2 β^2 + ... + a_{n?1} β^{n?1}

Suppose F ? K a finite extension. Then ?β∈K, ?f∈F[x] s.t.?β is a root of f.

Suppose F ? K field extension,?β∈K. Then ? a unique, monic, irreducible polynomial p ∈ F[x] having β as a root & if f ∈ F[x] is any polynomial w/ β as a root, then p|f. p is called the minimal polynomial of f over F.

Suppose F ? K a field extension f ∈ F[x], β∈K a root. Then ?σ∈Aut(K,F), σ(β) is also a root of f.

If F ? K a finite extension then Aut(K,F) is finite.

Fundamental Theorem of Galois Theory: Suppose K is a Galois extension of F, G = Aut(K,F). Then?there is a bijection {L ? K} ←→ {H < G}, L → L′ = H′ →?H = L:

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