50 lessons in Abstract Algebra

What is a group?Subgroups & Lagrange's theoremPermutations & the symmetric groupParity, sign & the alternating groupProof of Lagrange's theoremGroup homomorphismsMatrix groups: GL, SL, O and SONormal subgroupsQuotient groupsProof of first isomorphism theoremGroup actions & orbit-stabilizerThe isomorphism theoremsProof of orbit-stabilizerRotations of the tetrahedron and the cubeBurnside's counting lemmaConjugacy classes & the class equationCauchy's theorem & p-groupsStructure of finitely generated abelian groupsSylow's theoremsUsing Sylow: cyclic orders and groups that cannot be simpleEvery group of order up to 8RingsIdeals & quotient ringsPIDs, UFDs, Euclidean domainsPolynomial ringsFields & Galois previewField extensions & finite fieldsGalois correspondenceSemidirect productsComposition series & Jordan–HölderSolvable & nilpotent groupsSimple groups & classificationTransitivity & primitivitySimplicity of the alternating groupSolvability by radicals & Abel–RuffiniFree groups & presentationsFree products & amalgamationCayley graphsWreath productsGroup extensions & the Schur multiplierVector spaces as modulesTensor products of modulesModules over a PID & canonical formsSmith normal formCharacter theory & Schur orthogonalityGroup representations introSemisimple rings & Wedderburn–ArtinReflection & Coxeter groupsQuaternions, octonions & the division-algebra towerClifford algebras & the geometric product

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Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Abstract Algebra exactly where it's useful.

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