Group theory MOC

Group action

A group action¹ is a way to associate symmetries on a set (as automorphisms) with a group. group If 𝐺 is a group and Ω is a set², then a left group action is a map 𝛼 :𝐺 ×Ω →Ω :(𝑔,𝜔) ↦𝑔𝜔³ satisfying

1. Identity: 𝑒𝜔 =𝜔 for all 𝑚 ∈𝑀
2. Compatibility: 𝑔(ℎ𝜔) =(𝑔ℎ)𝜔 for all 𝑔,ℎ ∈𝐺 and 𝜔 ∈Ω.

and a right group action is a map 𝛽 :Ω ×𝐺 →Ω :(𝜔,𝑔) ↦𝜔𝑔 satisfying

1. Identity: 𝜔𝑒 =𝜔 for all 𝜔 ∈Ω
2. Compatibility: (𝜔𝑔)ℎ =𝜔𝑔ℎ for all 𝑔,ℎ ∈𝐺 and 𝜔 ∈Ω

The group 𝐺 is said to act on the space or structure Ω, where the function 𝛼(𝑔, −) is said to be the action of 𝑔 on Ω — which is always an automorphism. Ω is thence called a 𝐺-space.

Terminology

• A group actions associates to each point 𝜔 ∈Ω an orbit.
• For a given point 𝜔 ∈Ω, the set of group elements that map 𝑚 to itself are called the Stabilizer group, which is a subgroup.
• The set of all orbits is called the Orbit space or quotient.
• Types of action
  • Iff every stabilizer is {𝑒} the action is free.
  • Iff 𝛼(−)(𝜔) :𝐺 →Ω is surjective for all/any 𝜔 ∈Ω the action is transitive.
  • Iff 𝛼(−) :𝐺 →Aut⁡(Ω) is a group monomorphism the action is effective or faithful.
  • A Regular group action is free and transitive.
• The degree of 𝛼 is the cardinality of Ω.

Properties

• The product of the cardinality of the orbit and the order of the stabiliser is the order of the group (Orbit-stabilizer theorem)
• (Left-)𝐺-spaces form a Category of G-spaces with equivariant maps as morphisms.

Related concepts

• For topological properties, see Continuous group action
• Permutation group

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tidy | en | SemBr

Footnotes

1. German Wirkung or Operation. ↩

2. Usually taken to be a Space or an algebraic structure. ↩

3. When the action is understood the convention is to juxtapose the group element to the point/element in the set ↩