Find a concept
Start typing to search the mathematical index.
The orbits of a group action form a partition of the underlying set.
Orbit decomposition lemma. Let be a group acting on a set . Then the distinct orbits form a partition of .
Define when for some . This 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.
In axiomatic set theory, set and the membership relation are primitive notions. The axiom of extensionality determines equality by membership: 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:
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