抽象代数 Abstract Algebra 学习笔记

课本：

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

本笔记包含了课件里几乎所有知识点，一共13个week

Week1

概念

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

集合运算：

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, $\mathbf{{\color{Red} \sigma ^{-1}(C)}}$ = {a∈A | σ(a)∈C}
fiber of b= $\sigma ^{-1}(b)=\sigma ^{-1}(\left \{ b \right \})$ = {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

单射 injective(one-to-one): ∀a1, a2∈A, if σ(a1) = σ(a2) then a1 = a2
满射 surjective(onto): ∀b∈B, ∃a∈A s.t. σ(a) = b
双射 bijective: if it’s surjective & injective

identity function on A: ${\color{Red} \mathbf{id_A}}$ : A→A is defined by $\mathbf{id_A(a)= a}$ for all a∈A

inverse $\mathbf{{\color{Red} \sigma ^{-1}}}$ :
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, $\sigma ^{-1}$ = τ

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

Sets of Functions $\mathbf{{\color{Red} B^A}}$ = {σ : A→B | σ a function}, A, B are sets, 如C(R,R)={continuous functions σ: R→R} ⊂ $R^R$

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 = ∅
$\bigcup_{A\in \Omega}A=X$

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

对称群 the symmetric group of X / the permutation group of X Sym(X) = {f: X→X | f is a bijection} ⊂ $X^X$ , X nonempty set, 即Sym(X) are bijections from X to itself.
When X = {1, ..., n}, n∈Z, write $\mathbf{{\color{Red}S_n }}$ = Sym(X) symmetric group on n elements, 如S3 = Sym({1, 2, 3}), 有n!个元素
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
$\sigma ^0=id_X$
$\sigma ^{-1}=inverse, \sigma ^{-r}=(\sigma ^{-1})^r$ , r>0
r,s∈Z, $\sigma^{r+s}=\sigma^{r}\circ \sigma^{s}=\sigma^{s}\circ \sigma^{r}$

性质

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
$id_X$ ∈Sym(X), $id_X$ ∘ σ = σ ∘ $id_X$ = σ
∀σ∈Sym(X), $\sigma ^{-1}$ ∈Sym(X), σ is a 置换 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).

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

互质 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+

同余类 congruence class of a mod n $\mathbf{{\color{Red} [a]_n}}$ =[a]=equivalence of a modulo n = {b∈Z | b ≡ a mod n}
[a] = {a + kn | k∈Z}

$\mathbf{{\color{Red} Z_n}}$ ={ $[a]_n$ | a∈Z}, 即integers mod n
∀a∈ $Z_n$ , there are exactly n congruence classes modulo n: $Z_n$ = {[0], [1], [2], ..., [n-1]}

Fix n∈Z. Define + & · on $Z_n$

[a] + [b] = [a + b]
[a][b] = [ab]

Addition and Multiplication Tables:

Say [a] ∈ $Z_n$ is a unit or is invertible if ∃ [b] ∈ $Z_n$ so that [a][b] = [1].
inverse of [a] $\mathbf{{\color{Red} [a]^{-1}}}$ = [b] is unique

n≥2, $\mathbf{{\color{Red} Z^\times_n}}$ = {[a] ∈ $Z_n$ | [a] is a unit}
∀n≥2, $\mathbf{Z^\times_n}$ = {[a] ∈ $\mathbf{Z_n}$ | gcd(a, n)=1}
If p≥2 is prime, $Z^\times_n$ = {[1],...,[p-1]}, $|Z^\times_n|$ =p-1

Euler phi-function φ(n)= $|Z^\times_n|$

Complex Numbers:

addition & multiplication are commutative and associative
复共轭性 Complex conjugation: z = a + bi, z¯ = a - bi
Absolute value: |z| = √(a^2 + b^2)
复平面 complex plane
-z = -a - bi is the additive inverse of z = a + bi
∀ z ∈ C\{0}, the multiplicative inverse is $z^{-1}=\frac{1}{z}=\frac{\bar{z}}{|z|^2}=\frac{1}{|z|^2}\bar{z}$

性质

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 $Z_n$

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

复数 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) = $\boldsymbol{\mathbf{}|z|e^{i\theta }}$
- 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 $\boldsymbol{\mathbf{}a^{-1}}=\frac{1}{a}\in F: a\cdot \frac{1}{a}=1$

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

多项式 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 $\{{a_k}\}^{\infty}_{k=0}$ 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

性质

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 二次多项式 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

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

向量空间 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

线性变换 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)

性质

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 $A_{ij}$ are defined by $T(v_j) = A_{1j} w_1 + A_{2j} w_2 + ... + A_{mj} w_m$

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阶方阵 $\mathbf{{\color{Red} M_{n \times n}(F)}}$ : is also a ring

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

正交群 Orthogonal Group O(n) ={A ∈ GL(n, F) | $A^tA$ =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

交换群/阿贝尔群 Abelian group: * is commutative

二面体群 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

性质

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) = { $\phi_{A,v}$ | 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 (都很容易证明)

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

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

循环群 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：

性质

Let G be a group, let ℋ 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 ${\color{Red} \boldsymbol{\mathbf{}C_g (h)}}$ = {g∈G | gh = hg} < G
Z(G) =

群同态 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

正规子群 Normal Subgroup H◁G: A subgroup H < G is normal in G if ∀g∈G, gHg^-1 = H, 即gH=Hg.

群同构 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

自同构群 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 ${\color{Red} \mathbf{c_g}}$ : G→G, given by $\mathbf{c_g (h)= ghg^{-1}}$ .
$c_g$ ∈ Aut(G), $\mathbf{c^{-1}_g = c_{g^{-1}}}$

${\color{Red} \mathbf{L_g}}$ : G→G, $\mathbf{L_g (h) = gh}$ , G be a group, g∈G

性质

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)

二面体群 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

抽代杂谈(13): 二面体群 - 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

性质

Every permutation is a composition of 2-cycles
If σ ∈ S_n is a composition of k 2-cycles, ε(σ)= $(-1)^k$
If σ ∈ S_n is a k-cycle, then ε(σ)= $(-1)^{k-1}$ . ∵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
$g^{-1}_1 g_2$ ∈ 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|
可用于证明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

环同态 ring homomorphism: 比群的同态要求更高

性质

(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 $\tilde{\Phi}$ : G/N → Φ(G) < H s.t. $\tilde{\Phi}$ 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, ∴易证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, $G=N_1N_2\cdots N_k$ , ∀i = 1, ..., k, $(N_1N_2 \cdots N_{i-1} N_{i+1} \cdots N_k)\cap N_i$ = {e}, then G ≌ N1×N2×···×N_k.
N1, N2, ..., N_k ◁ G, $G=N_1N_2\cdots N_k$ , ∀i = 1, ..., k, $(N_1N_2 \cdots N_{i-1} N_{i+1} \cdots N_k)\cap N_i$ = {e}, then Φ: N1×N2×···×N_k → G is an isomorphism.

Week10

概念

半直积 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

等价定义：∀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}.

性质

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:

性质

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}
${\color{Red} \mathbf{G^+_4}}$ < G4, $G^+_4$ = {Φ∈G4 | Φ preserves orientation} = ker(det)⋂G4

Ru ≅ ( $Z^7_3$ × $Z^{11}_2$ ) ⋊ ((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.

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

Q̃(R) = {(a,b)∣a,b∈R, b≠0}, suppose R is an integral domain.
Define ∼ on Q̃(R) by (a,b) ∼ (a',b') ⇒ ab' = a'b. This is an equivalence relation.
Q(R) = equivalences of Q̃(R) w.r.t. ∼, and the equivalence class of (a, b) denoted ${\color{Red} \mathbf{\frac{a}{b}}}$ .
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.

性质

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 ${\color{Red} \mathbf{n_p (G)}}$ satisfies 2 conditions: $\mathbf{n_p (G)}$ ≡ 1 mod p, $\mathbf{n_p (G)}$ | |G|/p^k

If G is a finite group, P < G a Sylow p-subgroup. Then $\mathbf{n_p (G)}$ = 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 ≅ $Z_{pq}$
q∣p − 1: then G is isomorphic to one of 2 groups

For any prime p ≥ 2, $\mathbf{Z^\times_p\cong Z_{p-1}}$

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):
$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$
$\left ( \frac{a}{b} \right )\left ( \frac{c}{d} \right )=\frac{ac}{bd}$
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 ${\color{Red} \mathbf{dim_F K}}$ = [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}

性质

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: