Abstract Algebra

Groups, rings, and fields — a Dummit & Foote arc from the group axioms to Galois theory.

Group Theory

Groups & Group Actions — Closure, identity, inverses — and how a group permutes a set.
Cyclic & Dihedral Groups — C_n rotates, D_n adds mirrors — the symmetry groups behind orbit counting.
Symmetric Groups — S_n: permutations, cycle order = lcm, and parity.
Subgroups & Lagrange — |H| divides |G|; index [G:H] = |G|/|H|.
Cosets & Normal Subgroups — Cosets partition G; normal means gNg^-1 = N.
Quotient Groups — Cosets of a normal N form G/N, of order |G|/|N|.
Homomorphisms & Iso Theorems — G/ im .
The Class Equation — |G| = |Z(G)| + ∑ [G : C_G(x_i)].
Sylow Theorems — n_p 1 p and n_p | m.
Finite Abelian Groups — Products of prime-power cyclics; count = partitions.
The Orbit–Stabilizer Theorem — |G| = | orbit(x)| · | stab(x)| — where a point can go, times what holds it still.
Burnside's Lemma — # orbits = 1/|G| _g |X^g| — average the fixed points.

Rings & Ideals — Two operations; ideals absorb products.
Integral Domains & Fields — No zero divisors; ℤ/nℤ a field if and only if n prime.
Polynomial Irreducibility — Rational root theorem & Eisenstein's criterion.
Euclidean Domains, PIDs & UFDs — Euclidean ⊆ PID ⊆ UFD.
Chinese Remainder Theorem — ℤ/mn ℤ/m × ℤ/n for coprime m, n.

Semidirect Products — Twist two groups by an automorphism action to build non-abelian ones.
Direct Products of Groups — Componentwise products, element orders, and the recognition theorem.
Solvable Groups — Derived series, abelian-quotient chains, and which groups are solvable.
Nilpotent Groups — Upper central series, nilpotency class, and direct products of Sylow subgroups.
Jordan–Hölder Theorem — Composition series exist and their simple factors are unique.
Modules over a Ring — Abelian group plus a scalar action: vector spaces and abelian groups unified.
Structure Theorem for Modules over a PID — Invariant factors and elementary divisors classify modules over a PID.
Rational Canonical Form — Invariant factors, companion blocks, and a field-independent canonical form.
Separable & Inseparable Extensions — Repeated roots, the derivative test, and perfect fields.
Solvability by Radicals — A polynomial is solvable by radicals if and only if its Galois group is solvable.
Free Groups & Presentations — Reduced words, the universal property, and groups from generators and relations.