Group theory MOC

Group action

A group action¹ is a way to associate symmetries on a set (as automorphisms) with a group. group If is a group and is a set², then a left group action is a map ³ satisfying

1. Identity: for all
2. Compatibility: for all and .

and a right group action is a map satisfying

1. Identity: for all
2. Compatibility: for all and

The group is said to act on the space or structure , where the function is said to be the action of on — which is always an automorphism. is thence called a -space.

Terminology

• A group actions associates to each point an orbit.
• For a given point , the set of group elements that map 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 the action is free.
  • Iff is surjective for all/any the action is transitive.
  • Iff is a group monomorphism the action is effective or faithful.
  • A Regular group action is free and transitive.
• The degree of is the cardinality of .

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-)-spaces form a Category of G-spaces with equivariant maps as morphisms.

Related concepts

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

---

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 ↩