Group action

Orbit

Given an action of a group on a set , the orbit¹ of a point is the set of points that may be mapped to when acted upon, i.e. #m/def/group

It follows the restriction of an action onto an orbit is transitive, and the induced subgroup of is called the transitive constituent.

Properties

• Orbit cardinality divides the finite order of the group element
• Orbit-stabilizer theorem
• Since orbits partition , one can form an orbit space .
• Iff the action is said to be transitive.

See also

• Group action orbital, and the more general -orbit.
• Group action suborbit

#state/tidy | #lang/en | #SemBr