Group Theory
Permutations as Bijections
Composition of Permutations
Definition of Sₙ
Structure and Properties
Order of Sₙ
Definition and Properties
Commutativity of Disjoint Cycles
Multiplication of Cycles
Converting between Notations
Calculation from Cycle Decomposition
Least Common Multiple Method
Definition of a Transposition
Every Permutation as a Product of Transpositions
Minimal Transposition Decompositions
Definition of Parity
Sign of a Permutation
Properties of Parity
Definition of Aₙ
Properties and Structure
Order of Aₙ
Generators of Aₙ
Statement and Proof
Embedding Groups into Symmetric Groups
Regular Representation
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6. Cyclic Groups
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8. Cosets and Lagrange's Theorem