Group action

A group action¹ is a way to associate symmetries on a set (as automorphisms) with a group. #m/def/group If $𝐺$ is a group and $Ω$ is a set², then a left group action is a map $𝛼 : 𝐺 \times Ω \to Ω : (𝑔, 𝜔) \mapsto 𝑔 𝜔$ ³ satisfying

Identity: $𝑒 𝜔 = 𝜔$ for all $𝑚 \in 𝑀$
Compatibility: $𝑔 (ℎ 𝜔) = (𝑔 ℎ) 𝜔$ for all $𝑔, ℎ \in 𝐺$ and $𝜔 \in Ω$ .

and a right group action is a map $𝛽 : Ω \times 𝐺 \to Ω : (𝜔, 𝑔) \mapsto 𝜔 𝑔$ satisfying

Identity: $𝜔 𝑒 = 𝜔$ for all $𝜔 \in Ω$
Compatibility: $(𝜔 𝑔) ℎ = 𝜔 𝑔 ℎ$ for all $𝑔, ℎ \in 𝐺$ and $𝜔 \in Ω$

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 $𝜔 \in Ω$ an orbit.
For a given point $𝜔 \in Ω$ , 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 $𝛼 (-) (𝜔) : 𝐺 \to Ω$ is surjective for all/any $𝜔 \in Ω$ the action is transitive.
- Iff $𝛼 (-) : 𝐺 \to A u t (Ω)$ 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 $𝐺 𝖲 𝖾 𝗍$ with equivariant maps as morphisms.

For topological properties, see Continuous group action
Permutation group

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

German Wirkung or Operation. ↩
Usually taken to be a Space or an algebraic structure. ↩
When the action is understood the convention is to juxtapose the group element to the point/element in the set ↩

Group action

Terminology

Properties

Related concepts

Footnotes