Group action

A group action assigns each group element a bijection of a set so that the identity acts trivially and products act by composition. It captures symmetry through transformations that preserve the group’s multiplication law.

Orbit: The set of points obtainable from one point by applying every element of a group action. Each orbit is one region of the set connected by the action.
Group: A set with an associative binary operation, an identity element, and an inverse for every element. Its multiplication determines how successive transformations must combine.
Burnside's lemma: A counting theorem stating that the number of orbits of a finite group action equals the average number of fixed points. It counts distinct configurations by averaging the points fixed by each symmetry.
Semigroup action: An action of a semigroup on a set that respects multiplication but need not assign invertible transformations. Unlike group actions, its transformations need not have inverses.
Stabilizer subgroup: The subgroup of elements that leave a specified point fixed under a group action. It measures the symmetries that act without moving a chosen point.
Set: A collection of distinct objects treated as a single mathematical object. An action needs points on which the group elements operate.
Pólya enumeration theorem: A theorem that counts colorings or other configurations up to the symmetries of a finite group. It extends orbit counting to systematically classify colored configurations.
Monoid action: An action of a monoid on a set that respects multiplication and lets the identity act trivially. A monoid can act without every acting element being invertible.
Kernel of a group action: The normal subgroup of group elements that act as the identity transformation on every point. It identifies which group elements are invisible in this particular action.
Permutation: A bijection from a set to itself. Each group element acts as a permutation of the underlying set.

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