# group action

## 1. group action on a space

Consider a space, like Euclidean space. Consider a group. Then, the group *acts* on the space if we can make a group homomorphism from the elements of the group to transformations on the space. For example consider the space of triangles, which lie in Euclidean space. Consider the transformations that rotates a triangle by 0, 60, 120 degrees. We can make a group homomorphism from \(C_3\) to these transformations.

## 2. group action on a mathematical structure

Consider a mathematical structure, like a vector space. Consider a group. Then, the group *acts* on the mathematical structure if we can make a group homomorphism from the elements of the group to automorphisms on the structure. Note: the automorphisms on the structure form their own group. For example, a "group action on a vector space" means that we are considering a homomorphism between a given group and the general linear group. This mapping is called a *representation*.