Orbit (Group action)

Description

The Orbit is the set of all points that can be reached by applying group action to a specific base point.

Definition by action diagram

Let $G$ be a Group and $G\to Perm(S)$ a Group action. Taking the generators of the group we can visualise the effect of the group action by connecting two points with an arrow if they are related by multiplication of a generator. The orbit then consists of the component of a point. This is called an Operationendiagramm.

Formal Definition

Let $G$ be a Group, $X$ a set and $G\times X\to X,(g,x)\mapsto g\cdot x$ a Group action For a given $x\in X$, $G(x)=\{g\cdots x:g\in G\}$ is called the Orbit of $x$.

Properties

Zerlegung

The Orbits give a partition of the set $X$