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Orbits of a group action form a partition of the underlying set
Orbit Decomposition Lemma: Let be a group acting on a set via a group action. Then the set of orbits is a partition of .
Equivalently, define a relation on by if there exists with . Then is an equivalence relation, and its equivalence classes are exactly the orbits.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
A set is an object for which statements of the form (read “ is an element of ”) are meaningful, and whose identity is determined by extensionality: for sets ,
Let a group action of a group on a set be given, written . For , the orbit of is the subset
A partition of a set is a set of subsets of (called blocks or parts) such that:
A relation from a set to a set is a set with
where is the Cartesian product. If , one often writes .
An equivalence relation on a set is a relation satisfying the following three properties for all :
An equivalence class of an element (with respect to an equivalence relation on ) is the set