Abstract Algebra Expert

Group Theory

Basic Definitions

Group (G, ·):
  Closure:     a,b ∈ G → a·b ∈ G
  Associativity: (a·b)·c = a·(b·c)
  Identity:    ∃e: a·e = e·a = a
  Inverses:    ∀a ∃a⁻¹: a·a⁻¹ = a⁻¹·a = e

Abelian (commutative): a·b = b·a for all a,b ∈ G

Order:
  |G| = order of group (number of elements)
  |a| = order of element a = smallest n>0: aⁿ = e

Examples:
  (ℤ,+): integers under addition (infinite, abelian)
  (ℤₙ,+): integers mod n (finite, abelian, cyclic)
  (ℤₙ*,·): units mod n (finite, abelian)
  Sₙ: symmetric group on n elements (non-abelian for n≥3)
  Aₙ: alternating group (even permutations), |Aₙ| = n!/2
  Dₙ: dihedral group (symmetries of regular n-gon), |Dₙ| = 2n
  GL(n,F): invertible n×n matrices over field F
  SL(n,F): matrices with determinant 1
  Quaternion group Q₈: {±1,±i,±j,±k}

Subgroups & Cosets

Subgroup H ≤ G:
  H nonempty, closed under operation and inverses
  One-step test: a,b ∈ H → ab⁻¹ ∈ H
  Two-step test: closed under · and ⁻¹

Lagrange's theorem:
  H ≤ G (G finite): |H| divides |G|
  |G| = |H| · [G:H]  ([G:H] = index = number of cosets)
  Corollary: |a| divides |G|, so aˡᴳˡ = e

Left cosets: aH = {ah: h∈H}
  Cosets partition G (equivalence classes)
  All cosets have same size |H|

Normal subgroup H ⊴ G:
  gHg⁻¹ = H for all g ∈ G
  Equivalently: left and right cosets coincide gH = Hg
  Examples: any subgroup of abelian group, center Z(G)
  Kernel of homomorphism is always normal

Quotient group G/H (H normal):
  Elements: left cosets {aH: a∈G}
  Operation: (aH)(bH) = (ab)H
  |G/H| = |G|/|H| (G finite)

Homomorphisms & Isomorphisms

Homomorphism φ: G → H:
  φ(ab) = φ(a)φ(b) for all a,b ∈ G
  Properties: φ(e_G) = e_H, φ(a⁻¹) = φ(a)⁻¹

Kernel: ker(φ) = {g∈G: φ(g) = e_H} ⊴ G  (always normal!)
Image: im(φ) = {φ(g): g∈G} ≤ H

Isomorphism: bijective homomorphism (G ≅ H)
Automorphism: isomorphism from G to itself

First Isomorphism Theorem:
  G/ker(φ) ≅ im(φ)
  Key tool for quotient groups!

Second Isomorphism Theorem:
  H ≤ G, N ⊴ G: HN/N ≅ H/(H∩N)

Third Isomorphism Theorem:
  N ⊴ M ⊴ G: (G/N)/(M/N) ≅ G/M

Correspondence theorem:
  φ: G→G/N: subgroups of G/N ↔ subgroups of G containing N

Cyclic Groups & Permutations

Cyclic group ⟨a⟩ = {aⁿ: n∈ℤ}
  Every subgroup of cyclic group is cyclic
  ℤₙ cyclic of order n, ℤ cyclic infinite
  ⟨a⟩ ≅ ℤₙ if |a|=n, ≅ ℤ if |a|=∞

Permutation groups:
  σ ∈ Sₙ: bijection {1,...,n}→{1,...,n}
  Cycle notation: (1 2 3) means 1→2→3→1
  Transposition: 2-cycle (i j)
  Every permutation = product of disjoint cycles (unique up to order)
  |σ| = lcm of cycle lengths
  Sign: sgn(σ) = (-1)^(inversions) = (-1)^(n-c) (c = number of cycles including fixed points)
  Even/odd permutation: sgn = +1/-1

Alternating group Aₙ:
  Even permutations, |Aₙ| = n!/2
  A₅ is simple (no normal subgroups) — smallest non-abelian simple group
  This is why degree 5 polynomial not solvable by radicals!

Sylow Theory

Sylow p-subgroup: subgroup of order pᵏ where pᵏ | |G| but p^(k+1) ∤ |G|

Sylow's Theorems (p prime, pᵏ | |G|):
  1st: Sylow p-subgroup exists
  2nd: All Sylow p-subgroups are conjugate (isomorphic)
  3rd: nₚ = number of Sylow p-subgroups
       nₚ ≡ 1 (mod p)
       nₚ | |G|/pᵏ

Applications:
  Classify groups of small order
  Prove group is not simple (show nₚ = 1 → Sylow subgroup normal)
  Example: |G|=15=3·5: n₃|5 and n₃≡1(mod 3) → n₃=1; n₅|3 and n₅≡1(mod 5) → n₅=1
    Both Sylow subgroups normal → G ≅ ℤ₁₅ (cyclic)

Ring Theory

Ring (R, +, ·):
  (R, +): abelian group
  (R, ·): associative, distributive over +
  Ring with unity: has multiplicative identity 1
  Commutative ring: ab = ba

Examples:
  ℤ, ℚ, ℝ, ℂ: number rings
  ℤₙ: integers mod n
  M_n(R): n×n matrices over R (non-commutative)
  R[x]: polynomial ring over R
  R[x,y]: polynomials in two variables
  ℤ[i]: Gaussian integers {a+bi: a,b∈ℤ}

Types of elements:
  Unit: has multiplicative inverse (a·b=1)
  Zero divisor: a≠0, ∃b≠0: ab=0
  Nilpotent: aⁿ=0 for some n
  Idempotent: a²=a
  Integral domain: commutative, unity, no zero divisors
  Field: commutative, unity, every nonzero element is unit

Ideals:
  Left ideal: RI ⊆ I (rI ⊆ I for all r)
  Right ideal: IR ⊆ I
  Two-sided ideal: left and right
  Kernel of ring homomorphism is always an ideal

Principal ideal: (a) = {ra: r∈R} = aR
  PID (principal ideal domain): integral domain, every ideal principal
  Examples: ℤ, F[x] (polynomial ring over field), ℤ[i]

Prime ideal P: ab∈P → a∈P or b∈P
  In commutative ring: prime ideal ↔ R/P integral domain

Maximal ideal M: no ideal strictly between M and R
  In commutative ring: maximal ↔ R/M is a field

First isomorphism theorem for rings:
  φ: R→S ring homomorphism: R/ker(φ) ≅ im(φ)

Chinese Remainder Theorem for rings:
  I,J coprime ideals (I+J=R): R/(I∩J) ≅ R/I × R/J

UFD (Unique Factorization Domain):
  Integral domain, every element = unit × product of irreducibles (unique)
  PID → UFD (but not conversely)
  Examples: ℤ[x] is UFD but not PID (since (2,x) not principal)

Field Theory

Field: commutative ring where every nonzero element is a unit
  Examples: ℚ, ℝ, ℂ, ℤₚ (p prime), ℚ(√2), 𝔽₂ₙ

Field extensions:
  F ⊆ K (K contains F as subfield)
  [K:F] = dimₐ(K) = degree of extension
  Tower law: [K:F] = [K:E][E:F]  for F⊆E⊆K

Algebraic elements:
  α algebraic over F: f(α)=0 for some f∈F[x]
  Minimal polynomial: monic irreducible poly of smallest degree
  [F(α):F] = deg(min poly)
  Transcendental: not algebraic (π and e are transcendental over ℚ)

Algebraic extensions:
  F(α): smallest field containing F and α
  If α algebraic, F(α) ≅ F[x]/(min poly of α)

Splitting field:
  Smallest extension where polynomial f splits into linear factors
  Exists and unique up to isomorphism

Algebraic closure:
  F̄: field where every polynomial has a root
  ℂ = algebraic closure of ℝ (Fundamental Theorem of Algebra)

Finite fields:
  Order = pⁿ (p prime, n≥1)
  All fields of order pⁿ are isomorphic → 𝔽_{pⁿ}
  Multiplicative group 𝔽_{pⁿ}* is cyclic
  Subfields: 𝔽_{pᵐ} ⊆ 𝔽_{pⁿ} ↔ m|n
  Frobenius automorphism: x↦xᵖ generates Gal(𝔽_{pⁿ}/𝔽_p) ≅ ℤₙ

Galois Theory

Galois group:
  Gal(K/F) = Aut_F(K) = field automorphisms fixing F
  |Gal(K/F)| = [K:F] for Galois extensions

Galois extension K/F:
  Normal (splits over F) AND separable (distinct roots)
  Equivalent: |Gal(K/F)| = [K:F]
  Examples: ℚ(√2,√3)/ℚ, splitting fields of separable polynomials
  Non-example: ℚ(∛2)/ℚ (not normal)

Fundamental Theorem of Galois Theory:
  For Galois extension K/F with G = Gal(K/F):
  Correspondence: {subgroups of G} ↔ {intermediate fields F⊆E⊆K}
  H ↦ K^H = {x∈K: σ(x)=x ∀σ∈H}  (fixed field)
  E ↦ Gal(K/E)  (automorphisms fixing E)
  Reverses inclusion: H₁≤H₂ ↔ K^H₁ ⊇ K^H₂
  [K:E] = |Gal(K/E)|, [E:F] = [G:Gal(K/E)]
  E/F Galois ↔ Gal(K/E) ⊴ G, and Gal(E/F) ≅ G/Gal(K/E)

Solvability by radicals:
  f(x) solvable by radicals ↔ Gal(f) is solvable group
  Group G solvable: G = G₀⊃G₁⊃...⊃Gₖ={e} with Gᵢ/Gᵢ₊₁ abelian
  A₅ is not solvable → general degree 5 polynomial not solvable!
  (Abel-Ruffini theorem)

Classical ruler-compass constructions:
  α constructible ↔ [ℚ(α):ℚ] = 2ⁿ
  Squaring circle: impossible (π transcendental)
  Doubling cube: impossible (∛2: degree 3, not power of 2)
  Trisecting angle: usually impossible
  Regular n-gon constructible ↔ n = 2ᵏ·p₁·p₂...pₘ (pᵢ Fermat primes)

Module Theory

Module M over ring R:
  (M, +): abelian group
  R acts on M: r·m ∈ M with distributivity and associativity
  Vector spaces: modules over a field

Submodule: subgroup closed under R-action
Quotient module: M/N for submodule N
Module homomorphism (R-linear map): f(rm) = rf(m)

Free module: M ≅ R^n (has basis)
  Finitely generated: spanned by finite set
  Free → finitely generated (converse fails over general rings)

Classification of modules over PIDs:
  Finitely generated module M over PID R:
  M ≅ R^r ⊕ R/(d₁) ⊕ R/(d₂) ⊕ ... ⊕ R/(dₖ)
  d₁|d₂|...|dₖ (invariant factors)
  Special case (R=ℤ): finitely generated abelian groups!

Exact sequences:
  0 → A →ᶠ B →ᵍ C → 0  (short exact sequence)
  Exact: im(f) = ker(g)
  Short exact: f injective, g surjective, im(f) = ker(g)
  Split: sequence is "isomorphic" to 0 → A → A⊕C → C → 0

Tensor product:
  M⊗ₐN: universal bilinear map
  R⊗_R M ≅ M
  Hom(M,N): module of R-linear maps

Projective/injective modules:
  Projective: direct summand of free module
  Injective: Hom(-,M) exact
  Flat: M⊗- exact

Representation Theory

Representation of group G:
  Homomorphism ρ: G → GL(V) for vector space V over field k
  Degree = dim(V)

Subrepresentation: V-subspace invariant under all ρ(g)
Irreducible (simple): no proper nonzero subrepresentation

Maschke's theorem:
  G finite, char(k) ∤ |G|: every representation is completely reducible
  V = V₁ ⊕ V₂ ⊕ ... ⊕ Vₖ (direct sum of irreducibles)

Character:
  χᵥ(g) = Tr(ρ(g)) (trace of representation matrix)
  Class function: χ(hgh⁻¹) = χ(g) (constant on conjugacy classes)
  Characters of irreps: orthogonal basis for class functions
  ⟨χ,ψ⟩ = (1/|G|) Σ χ(g)ψ(g)⁻ = δ_{irreps}

Number of irreps = number of conjugacy classes
Sum of squares of dimensions = |G|: Σ (dim Vᵢ)² = |G|

Character table:
  Rows: irreducible representations
  Columns: conjugacy classes
  Entry: character value χ(g)

Regular representation:
  G acts on k[G] by left multiplication
  Decomposes as direct sum of each irrep with multiplicity = degree

Category Theory Basics

Category C:
  Objects: collection ob(C)
  Morphisms: for each pair A,B: hom(A,B) (arrows A→B)
  Composition: f:A→B, g:B→C → g∘f:A→C (associative)
  Identities: 1_A: A→A for each A

Examples:
  Set: sets and functions
  Grp: groups and homomorphisms
  Ring: rings and ring homomorphisms
  Top: topological spaces and continuous maps
  Vect_k: vector spaces over k and linear maps

Functor F: C→D:
  Assigns object F(A)∈D to each A∈C
  Assigns morphism F(f) to each morphism f
  Preserves composition and identities
  Covariant: F(g∘f) = F(g)∘F(f)
  Contravariant: reverses arrows

Natural transformation η: F⟹G:
  For each A: η_A: F(A)→G(A) (natural in A)
  Commutes with morphisms

Universal properties:
  Products: A×B with projections π₁,π₂
  Coproducts: A+B with injections i₁,i₂
  Free objects: free group on set S
  Tensor products, kernels, cokernels

Adjoint functors:
  F⊣G: hom(F(A),B) ≅ hom(A,G(B)) (natural bijection)
  Free-forgetful adjunction: free group on S ⊣ underlying set

Common Pitfalls

Related Skills

• number-theory-expert: Algebraic number theory
• topology-expert: Algebraic topology uses groups extensively
• linear-algebra-expert: Modules generalize vector spaces
• calculus-expert: Lie groups connect algebra and analysis
• cryptography-expert: Groups and finite fields in crypto
• physics-quantum-mechanics: Group representations in physics