Group action

A group action is a function that exploits symmetries in a space using group properties.

Definition

A group action of on some object is a homomorphism into the endomorphism group of the object.

Equivalent (somewhat easier to use) definition

A (left) group action of a group on a set is a function that satisfies the following conditions: and

The set together with the action of on is called a G-set.

We can define a right action… but literally everything I have used it for uses left actions… so I’ll leave it for now.

Notation

Generally we denote action by putting the elements next to each other or with . For example, or

Morphism between -sets

If and are two -sets, a morphism from to is a function such that

Automorphism

An automorphism is a special group of morphisms from the -set to itself. These are denoted