Permutation Groups -

A is a set of bijections (one-to-one and onto mappings) from a set to itself that forms a group under the operation of function composition . These groups are fundamental in abstract algebra because they can represent the symmetries of geometric objects, like the rotations and reflections of a triangle or square. Core Concepts The Set ( ): Usually represented as . A permutation shuffles these elements. The Operation: We use composition ( ∘composed with ). Applying means first performing The Symmetric Group ( Sncap S sub n ): This is the group of all possible permutations of elements. Its size (order) is (n-factorial). Key Notations Abstract Algebra - 5.2 Permutation Groups