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group-theory

The group axioms

taught

Taught in

Requires

Required by

Local knowledge map

Local knowledge map

Cayley's theorem Consequence of Lagrange's theorem: element order divides… Cycle notation Cyclic groups Fixed points of a group element Group isomorphisms Image of a homomorphism Injective and surjective homomorphisms Inner automorphisms Internal direct products Intersections of subgroups Kernel of a homomorphism Left and right cosets Linear codes as subgroups of (F_2)^n Normal subgroups Orbits and stabilizers Order of elements in Z_n and U(n) Orders of elements in a direct product Powers and integer exponents of a group element Products of subgroups Subgroups generated by a subset Symmetry and conservation laws (Noether's theorem) The center of a group The conjugation action and conjugacy classes The finite subgroup test The group U(n) The normalizer of a subgroup p-groups Abelian groups Conjugation External direct products Group actions Group homomorphisms Motivating group actions: groups that move things Order of a group Order of an element Permutations as bijections Testing whether a structure is a group The Rubik's Cube as a group The group ℤₙ The subgroup test The group axioms Cayley tables for Dₙ Congruence modulo n The dihedral group Dₙ The symmetric group Sₙ Cayley (operation) tables Identity elements and inverses

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Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.