# Abstract Algebra

Abstract algebra studies operations stripped to their axioms. A
**group** is the leanest of them — a set with one
associative operation, an identity, and inverses — and this modest package
already captures the mathematics of _symmetry_.


_Figure 001 — A cyclic group: one generator sweeps out every element, ℤ/6 as rotations
of a hexagon.
_

_Figure 002 — Dihedral symmetry: rotations and reflections of a triangle, the full group
of its rigid motions.
_

The first examples are the symmetries of a shape. Rotate a polygon and it
lands on itself; reflect it and it does too. Compose those moves and you
get a group — cyclic when only rotations count, **dihedral**
once reflections join in.


Structure is enforced by **subgroups**. A subgroup sits inside
a group and splits it into _cosets_ — translated copies, all the
same size — so the order of any subgroup must divide the order of the
whole. That single counting fact, Lagrange's theorem, constrains what
groups can even exist.


_Figure 003 — Cosets partition a group into equal blocks — the idea behind Lagrange's
theorem and quotient groups.
_

To compare groups you use **homomorphisms**: maps that respect
the operation. Their reach is measured by the _image_, their
collapse by the _kernel_, and the two are tied together by the
isomorphism theorems — a quotient by the kernel is a faithful copy of the
image.


_Figure 004 — A homomorphism ℤ/6 → ℤ/3: source elements collapse onto their images, the
kernel mapping to the identity.
_

Layer a second operation on and you get **rings** and
**fields** — the axioms behind arithmetic. Polynomials,
integers, and matrices are all rings; fields are where division always
works, and their extensions are the setting for Galois theory.


The finite structures reveal themselves in their Cayley
tables. Every row and column is a permutation of the elements — a
Latin square — which is just invertibility made visible.


_Figure 005 — A Cayley table: the whole operation written out, each row and column a
permutation of the group.
_

The payoff is **Galois theory**, which pins field extensions
to groups of symmetries and settles classical questions — why the general
quintic has no formula in radicals, why some angles cannot be trisected by
compass and straightedge.


Learn the axioms once and the same skeleton appears everywhere: in number
theory, in geometry, in cryptography, in the symmetries of physics. Abstract
algebra is the grammar those subjects are written in.


---

## Contents

### 1. Foundations

1. [Sets, Functions, and Equivalence Relations](/abstract-algebra/foundations/sets-functions-relations)
2. [The Integers and Modular Arithmetic](/abstract-algebra/foundations/integers-and-modular-arithmetic)

### 2. Groups and Symmetry

1. [Group Axioms and First Examples](/abstract-algebra/groups-and-symmetry/group-axioms-and-first-examples)
2. [Dihedral and Symmetric Groups](/abstract-algebra/groups-and-symmetry/dihedral-and-symmetric-groups)
3. [Matrix and Quaternion Groups](/abstract-algebra/groups-and-symmetry/matrix-and-quaternion-groups)
4. [Homomorphisms, Isomorphisms, and Actions](/abstract-algebra/groups-and-symmetry/homomorphisms-and-group-actions)

### 3. Subgroups and Quotients

1. [Subgroups and Their Substructures](/abstract-algebra/subgroups-and-quotients/subgroups-and-substructures)
2. [Cyclic Groups](/abstract-algebra/subgroups-and-quotients/cyclic-groups)
3. [Generation and the Lattice of Subgroups](/abstract-algebra/subgroups-and-quotients/generation-and-subgroup-lattices)
4. [Cosets, Lagrange, and Normal Subgroups](/abstract-algebra/subgroups-and-quotients/cosets-lagrange-and-normal-subgroups)
5. [The Isomorphism Theorems](/abstract-algebra/subgroups-and-quotients/isomorphism-theorems)
6. [Composition Series and the Alternating Group](/abstract-algebra/subgroups-and-quotients/composition-series-and-the-alternating-group)

### 4. Group Actions and Sylow Theory

1. [Actions, Orbits, and Cayley's Theorem](/abstract-algebra/group-actions-and-sylow/actions-and-cayleys-theorem)
2. [Conjugation and the Class Equation](/abstract-algebra/group-actions-and-sylow/conjugation-and-the-class-equation)
3. [The Sylow Theorems](/abstract-algebra/group-actions-and-sylow/sylow-theorems)
4. [Automorphisms and Simplicity of Aₙ](/abstract-algebra/group-actions-and-sylow/automorphisms-and-simple-groups)

### 5. Products and Group Structure

1. [Direct Products and Finite Abelian Groups](/abstract-algebra/products-and-group-structure/direct-products-and-finite-abelian-groups)
2. [Semidirect Products](/abstract-algebra/products-and-group-structure/semidirect-products)
3. [p-Groups, Nilpotent, and Solvable Groups](/abstract-algebra/products-and-group-structure/nilpotent-and-solvable-groups)
4. [Classifying Groups of Small Order](/abstract-algebra/products-and-group-structure/classifying-small-groups)

### 6. Ring Theory

1. [Rings: Definitions and Examples](/abstract-algebra/ring-theory/rings-definitions-and-examples)
2. [Ideals, Quotient Rings, and Homomorphisms](/abstract-algebra/ring-theory/ideals-quotients-and-homomorphisms)
3. [Fields of Fractions and the CRT](/abstract-algebra/ring-theory/fractions-and-the-chinese-remainder-theorem)

### 7. Factorization and Polynomial Rings

1. [Euclidean Domains, PIDs, and UFDs](/abstract-algebra/factorization-and-polynomials/euclidean-domains-pids-ufds)
2. [Polynomial Rings over Fields](/abstract-algebra/factorization-and-polynomials/polynomial-rings-over-fields)
3. [Gauss's Lemma and Unique Factorization](/abstract-algebra/factorization-and-polynomials/gauss-lemma-and-unique-factorization)
4. [Irreducibility Criteria and Gröbner Bases](/abstract-algebra/factorization-and-polynomials/irreducibility-criteria-and-groebner)

### 8. Module Theory

1. [Introduction to Modules](/abstract-algebra/module-theory/introduction-to-modules)
2. [Generation, Direct Sums, and Free Modules](/abstract-algebra/module-theory/free-modules-and-direct-sums)
3. [Tensor Products and Exact Sequences](/abstract-algebra/module-theory/tensor-products-and-exact-sequences)
4. [Vector Spaces and Linear Maps](/abstract-algebra/module-theory/vector-spaces-and-linear-maps)

### 9. Modules over PIDs and Canonical Forms

1. [The Structure Theorem for Modules over a PID](/abstract-algebra/modules-over-pids/structure-theorem-over-pids)
2. [Rational Canonical Form](/abstract-algebra/modules-over-pids/rational-canonical-form)
3. [Jordan Canonical Form](/abstract-algebra/modules-over-pids/jordan-canonical-form)

### 10. Field Theory

1. [Field Extensions and Algebraic Elements](/abstract-algebra/field-theory/field-extensions-and-algebraic-elements)
2. [Straightedge-and-Compass Constructions](/abstract-algebra/field-theory/straightedge-and-compass-constructions)
3. [Splitting Fields and Algebraic Closure](/abstract-algebra/field-theory/splitting-fields-and-algebraic-closure)
4. [Separable Extensions and Cyclotomic Fields](/abstract-algebra/field-theory/separable-and-cyclotomic-extensions)

### 11. Galois Theory

1. [The Galois Correspondence](/abstract-algebra/galois-theory/the-galois-correspondence)
2. [Finite Fields](/abstract-algebra/galois-theory/finite-fields)
3. [Cyclotomic and Abelian Extensions](/abstract-algebra/galois-theory/cyclotomic-and-abelian-extensions)
4. [Galois Groups of Polynomials](/abstract-algebra/galois-theory/galois-groups-of-polynomials)
5. [Solvability by Radicals and the Quintic](/abstract-algebra/galois-theory/solvability-by-radicals-and-the-quintic)

### 12. Capstone: Where Algebra Goes Next

1. [A Glimpse of Commutative Algebra and Algebraic Geometry](/abstract-algebra/capstone/commutative-algebra-and-algebraic-geometry)
2. [A Glimpse of Representation and Character Theory](/abstract-algebra/capstone/representation-and-character-theory)