Abstract Algebra: Structural Mathematics

Last updated 12 Jan 2026, 14:40

Abstract algebra studies algebraic structures: groups, rings, fields, modules, and their homomorphisms. It provides the foundation for modern number theory, geometry, and cryptography.

Group Theory

Definition

A group (G, ·) satisfies:

  1. Closure: a·b ∈ G
  2. Associativity: (a·b)·c = a·(b·c)
  3. Identity: ∃e: e·a = a·e = a
  4. Inverses: ∀a ∃a⁻¹: a·a⁻¹ = e

Abelian/commutative: a·b = b·a

Examples

  • (ℤ, +): Integers under addition
  • (ℤ/nℤ, +): Integers mod n
  • (ℤ/pℤ)×: Multiplicative group, p prime
  • Sₙ: Symmetric group on n elements
  • GLₙ(F): Invertible n×n matrices
  • Dₙ: Dihedral group (symmetries of n-gon)

Subgroups

H ≤ G if H is a group under G's operation.

Subgroup test: H ≤ G ⟺ H ≠ ∅ and ab⁻¹ ∈ H for all a,b ∈ H.

Lagrange's theorem: |H| divides |G|.

Cosets and Quotients

Left coset: aH = {ah : h ∈ H} Index [G:H] = number of cosets

Normal subgroup: N ◁ G if gNg⁻¹ = N for all g.

Quotient group: G/N with operation (aN)(bN) = (ab)N.

Homomorphisms

φ: G → H with φ(ab) = φ(a)φ(b).

  • Kernel: ker φ = φ⁻¹(e_H), always normal
  • Image: im φ = φ(G), subgroup of H

First isomorphism theorem: G/ker φ ≅ im φ

Group Actions

Action of G on X: G × X → X with:

  • e·x = x
  • (gh)·x = g·(h·x)

Orbit-stabilizer: |G| = |Orb(x)| · |Stab(x)|

Sylow Theorems

For |G| = p^n·m with gcd(p,m) = 1:

  1. Sylow p-subgroups of order p^n exist
  2. All Sylow p-subgroups are conjugate
  3. Number of Sylow p-subgroups ≡ 1 (mod p) and divides m

Classification

  • Cyclic groups: ℤₙ
  • Abelian groups: Products of cyclic groups (fundamental theorem)
  • Simple groups: No proper normal subgroups
  • Classification of finite simple groups: Cyclic, alternating, Lie type, sporadic

Ring Theory

Definition

A ring (R, +, ·) satisfies:

  1. (R, +) is an abelian group
  2. · is associative
  3. Distributivity: a(b+c) = ab+ac, (a+b)c = ac+bc

Commutative ring: ab = ba Ring with identity: ∃1: 1·a = a·1 = a

Examples

  • ℤ: Integers
  • ℤ[x]: Polynomials with integer coefficients
  • ℤ[i]: Gaussian integers
  • Mₙ(R): n×n matrices
  • ℤ/nℤ: Integers mod n

Ideals

I ⊆ R is an ideal if:

  • I is an additive subgroup
  • ra ∈ I and ar ∈ I for all r ∈ R, a ∈ I

Principal ideal: (a) = Ra = {ra : r ∈ R} Prime ideal: ab ∈ P ⟹ a ∈ P or b ∈ P Maximal ideal: No ideal properly between M and R

Quotient Rings

R/I with (a+I)·(b+I) = (ab+I).

  • R/P is integral domain ⟺ P is prime
  • R/M is field ⟺ M is maximal

Ring Homomorphisms

φ: R → S preserving both operations.

First isomorphism theorem: R/ker φ ≅ im φ

Special Rings

  • Integral domain: No zero divisors (ab = 0 ⟹ a = 0 or b = 0)
  • Principal ideal domain (PID): Every ideal is principal
  • Unique factorization domain (UFD): Elements factor uniquely into irreducibles
  • Euclidean domain: Has division algorithm

Hierarchy: Fields ⊂ Euclidean ⊂ PID ⊂ UFD ⊂ Integral domains

Polynomial Rings

R[x] = polynomials with coefficients in R.

Gauss's lemma: ℤ[x] is a UFD. Eisenstein's criterion: Irreducibility test. Hilbert basis theorem: R Noetherian ⟹ R[x] Noetherian.

Field Theory

Definition

Field: Commutative ring where every nonzero element has multiplicative inverse.

Examples

  • ℚ, ℝ, ℂ: Rationals, reals, complex numbers
  • ℤ/pℤ = 𝔽_p: Finite field of p elements
  • ℚ(√2): Rationals with √2 adjoined
  • 𝔽_q: Finite field of q = p^n elements (unique up to isomorphism)

Field Extensions

K/F: F is a subfield of K.

Degree: [K:F] = dim_F K Tower law: [L:F] = [L:K]·[K:F]

Algebraic element: α satisfies polynomial with F-coefficients. Transcendental: Not algebraic (e.g., π, e over ℚ).

Splitting Fields

Smallest extension where polynomial factors completely.

Exists and unique up to isomorphism.

Galois Theory

For Galois extension K/F:

Galois group: Gal(K/F) = Aut_F(K)

Fundamental theorem:

  • Subgroups of Gal(K/F) ↔ Intermediate fields F ⊆ E ⊆ K
  • [K:E] = |H| where H corresponds to E
  • Normal subgroups ↔ Galois extensions

Applications

  • Impossibility of quintic formula (A₅ is not solvable)
  • Impossibility of ruler-and-compass constructions
  • Fundamental theorem of algebra

Module Theory

Definition

R-module M: Abelian group with R-action satisfying:

  • r(m+n) = rm + rn
  • (r+s)m = rm + sm
  • (rs)m = r(sm)
  • 1m = m

Generalizes vector spaces (R = field) and abelian groups (R = ℤ).

Submodules and Quotients

Analogous to subgroups and quotient groups.

Free Modules

R^n: Has basis, every element uniquely expressible.

Not all modules are free (torsion).

Structure Theorem

Finitely generated modules over PID: M ≅ R^r ⊕ R/(d₁) ⊕ ... ⊕ R/(d_k)

where d₁|d₂|...|d_k (invariant factors).

Tensor Products

M ⊗_R N: Universal object for bilinear maps.

Properties:

  • R ⊗_R M ≅ M
  • (M ⊕ N) ⊗ P ≅ (M ⊗ P) ⊕ (N ⊗ P)
  • Change of rings: S ⊗_R M

Representation Theory

Group Representations

ρ: G → GL(V): Homomorphism to linear automorphisms.

Character: χ_ρ(g) = tr(ρ(g))

Irreducible Representations

No proper invariant subspaces.

Maschke's theorem: Representations of finite groups over ℂ decompose into irreducibles.

Character Tables

  • Rows: Irreducible characters
  • Columns: Conjugacy classes
  • Orthogonality relations

Applications

Cryptography

  • RSA: ℤ/nℤ for n = pq
  • Elliptic curves: Group law for key exchange
  • AES: Finite field arithmetic in GF(2⁸)

Coding Theory

  • Linear codes: Subspaces of 𝔽_q^n
  • Cyclic codes: Ideals in 𝔽_q[x]/(x^n-1)
  • Reed-Solomon: Evaluation of polynomials

Physics

  • Symmetry groups in quantum mechanics
  • Lie groups and algebras
  • Representation theory for particle physics