Group action

Orbit

Given an action of a group 𝐺 on a set Ω, the orbit¹ 𝐺𝜔 of a point 𝜔 ∈Ω is the set of points that 𝑚 may be mapped to when acted upon, i.e. group

𝐺Ω={𝑔𝜔:𝜔∈Ω}

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

Properties

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

See also

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

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Footnotes

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