Mathematics / core
Groups and symmetry
Develop algebraic symmetry from concrete permutations to quotient groups, group actions, and structure theorems.
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Inside this module
Module 01 core
Groups and symmetry
Develop algebraic symmetry from concrete permutations to quotient groups, group actions, and structure theorems.
- 01
Symmetry operations and the group axioms
Recognize reversible transformations as a group and distinguish composition order from the shape being transformed.
25 min - 02
Subgroups and cyclic groups
Build groups inside groups, recognize generators, and calculate element orders without confusing them with group size.
26 min - 03
Permutation groups and parity
Calculate permutations in cycle notation and prove that their parity is independent of a chosen transposition decomposition.
28 min - 04
Cosets and Lagrange's theorem
Partition a group into equal-sized translates and distinguish necessary divisibility conditions from unjustified converse claims.
27 min - 05
Homomorphisms and normal subgroups
Track structure-preserving maps through kernels and images, then see why normality is the right condition for forming quotient groups.
28 min - 06
Quotient groups and isomorphism theorems
Build a group of cosets, prove why normality is necessary, and interpret a homomorphism through the information it discards.
27 min - 07
Group actions and orbit–stabilizer
Turn abstract group elements into permutations of a set and use stabilizers to count reachable configurations.
27 min - 08
Conjugacy classes and the class equation
Use the conjugation action to organize a finite group and prove why finite prime-power groups have nontrivial centers.
27 min - 09
Sylow Theorems and Finite Group Structure
Use prime-power subgroups, conjugation, and counting to prove normality and determine what a group order really forces.
28 min - 10
Finitely Generated Abelian Groups
Separate free directions from finite torsion, translate invariant factors into elementary divisors, and classify abelian groups of order 72.
28 min