Group Theory

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• What is abstract algebra? ~12 min
• Symmetry as motivation ~14 min
• Binary operations and closure ~14 min
• Associativity ~12 min
• Identity and inverses ~14 min
• Cayley tables ~12 min
• The group axioms ~12 min
• Groups and non-groups ~14 min
• A first gallery of groups ~15 min

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• Abelian vs. non-abelian groups ~12 min
• Finite and infinite groups; order of a group ~12 min
• Order of an element ~14 min
• Powers and integer exponents of an element ~13 min
• Cyclic groups ~13 min
• Generators of a cyclic group ~12 min
• Subgroups: definition and first examples ~14 min
• The trivial subgroup and generated subgroups ~13 min

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• Congruence modulo n, revisited as group structure ~14 min
• The group Z_n under addition ~13 min
• Units modulo n and the group U(n) ~14 min
• The greatest common divisor and group structure ~16 min
• Euler's phi function ~13 min
• Generators of Z_n ~12 min
• Order of elements in Z_n and U(n) ~13 min
• Classifying finite cyclic groups ~14 min
• A first look at modular exponentiation ~15 min

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• Permutations as bijections and two-line notation ~16 min
• The symmetric group Sₙ ~15 min
• Cycle notation ~16 min
• Disjoint cycle decomposition ~17 min
• Composing permutations ~17 min
• Inverse permutations ~15 min
• Transpositions and generating Sₙ ~18 min
• Parity -- even and odd permutations ~18 min
• The alternating group Aₙ ~17 min

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• Symmetries of geometric objects, revisited ~14 min
• The dihedral group Dₙ ~15 min
• Rotations and reflections as generators ~16 min
• Composing symmetries and non-abelian behavior ~16 min
• A presentation of Dₙ ~17 min
• Cayley tables for Dₙ ~16 min
• Symmetry groups of regular polygons ~15 min
• Dₙ as a subgroup of Sₙ ~17 min

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• The subgroup test ~15 min
• The finite subgroup test ~14 min
• Cyclic subgroups, revisited ~15 min
• Subgroups generated by a subset ~15 min
• Intersections of subgroups ~14 min
• Products of subgroups ~16 min
• Subgroups of cyclic groups ~15 min
• The subgroup lattice ~15 min

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• Left and right cosets ~16 min
• Coset representatives and the coset partition ~15 min
• The index of a subgroup ~13 min
• Lagrange's theorem ~15 min
• Consequences: element order divides group order ~12 min
• Groups of prime order are cyclic ~12 min
• Possible orders of elements and subgroups ~12 min
• Euler's theorem and Fermat's little theorem ~18 min

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• When left and right cosets agree ~16 min
• Conjugation ~16 min
• The definition of a normal subgroup ~18 min
• Normality tests ~17 min
• The center of a group ~17 min
• The normalizer of a subgroup ~17 min
• Normal subgroups: examples and counterexamples ~18 min

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• Motivation: collapsing cosets into elements ~15 min
• Why normality makes the operation well-defined ~18 min
• The quotient group G/N ~18 min
• Quotient groups of cyclic groups ~16 min
• Quotient groups of permutation groups ~17 min
• A first look at the First Isomorphism Theorem ~16 min
• The quotient construction as a recurring pattern ~14 min

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• Group homomorphisms: definition and examples ~18 min
• The kernel of a homomorphism ~20 min
• The image of a homomorphism ~18 min
• Injective and surjective homomorphisms ~20 min
• Isomorphisms ~20 min
• Automorphisms ~18 min
• The First Isomorphism Theorem ~24 min
• The Second Isomorphism Theorem ~22 min
• The Third Isomorphism Theorem ~20 min
• The Correspondence Theorem ~22 min

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• The external direct product ~16 min
• Orders of elements in a direct product ~15 min
• When is a direct product cyclic? ~16 min
• Internal direct products ~17 min
• Recognizing an internal direct product ~15 min
• The classification of finite abelian groups ~18 min
• Applying the classification theorem ~16 min
• A first look at semidirect products ~17 min

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• Groups that move things ~13 min
• The definition of a group action ~15 min
• Orbits ~15 min
• Stabilizers ~15 min
• The orbit-stabilizer theorem ~17 min
• Transitive and faithful actions ~15 min
• The conjugation action and conjugacy classes ~16 min
• Cayley's theorem revisited ~16 min

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• Fixed points of a group element ~14 min
• Burnside's lemma ~18 min
• Counting distinct colorings ~15 min
• Necklace and bracelet problems ~16 min
• A first look at Pólya enumeration ~15 min
• Symmetry and combinatorial counting ~15 min
• Choosing a group action to solve a counting problem ~16 min

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• Groups of prime order ~14 min
• Groups of order p² ~18 min
• Groups of order pq ~17 min
• Small-order classification as a worked recipe ~20 min
• Finite abelian groups, revisited ~15 min
• Simple groups: a first look ~16 min
• Composition series ~18 min
• The Jordan–Hölder theorem ~16 min

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• p-groups ~16 min
• Sylow p-subgroups: definition ~17 min
• The First Sylow Theorem: existence ~20 min
• The Second Sylow Theorem: conjugacy ~19 min
• The Third Sylow Theorem: counting ~20 min
• Using Sylow's theorems to classify groups ~21 min
• Normal Sylow subgroups ~18 min

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• Generators and relations ~18 min
• Free groups ~16 min
• Finitely generated and finitely presented groups ~17 min
• Group presentations revisited ~19 min
• The commutator and the commutator subgroup ~18 min
• Abelianization ~17 min
• Solvable groups ~19 min
• Nilpotent groups ~17 min
• Inner and outer automorphisms ~18 min
• Characteristic subgroups ~16 min
• Group extensions and the Schur–Zassenhaus theorem ~17 min

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• What a Cayley graph is ~18 min
• Generating sets and Cayley graphs ~18 min
• Directed versus undirected Cayley graphs ~18 min
• Reading group structure from a graph ~20 min
• Group theory and graph theory: connections ~20 min

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• Cryptography: RSA, completed ~22 min
• Algebraic coding theory ~22 min
• Symmetry in geometry: frieze and wallpaper groups ~16 min
• Molecular symmetry and crystallography ~16 min
• The Rubik's Cube as a group ~14 min
• Group theory in physics: symmetry and conservation laws ~14 min
• Group theory in computer science and coding ~14 min

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• Synthesizing group identification ~16 min
• Synthesizing subgroup and coset computation ~18 min
• Synthesizing homomorphisms and isomorphisms ~20 min
• Synthesizing group actions and counting ~18 min
• Synthesizing classification and Sylow ~20 min
• The group investigation: a guided case study ~20 min