Orbit (Group)

definitiongroup-theory

Definition

The orbit of $x$ with respect to a group $G$ is the set of all elements equal to the possible actions on $x$ with elements $g\in G$.

$$\mathrm{orb}(x)={\mathrm{orb}}_G(x)=\{g\bullet x:g\in G\}$$

Examples

1. $G=G{L}_2(\mathbb{R})\circlearrowleft {\mathbb{R}}^2$ $A\bullet v=Av$ If $v=\left(\begin{array}{c}0\\ 0\end{array}\right)$ $\mathrm{orb}(v)=\{\left(\begin{array}{c}0\\ 0\end{array}\right):A\left(\begin{array}{c}0\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)\}$ We need $A$ invertible, first column is $\left(\begin{array}{c}a\\ b\end{array}\right)$. For instance, $A=\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ will work. $A^{-1}=\frac{1}{a^2+b^2}\left(\begin{array}{cc}a& b\\ -b& a\end{array}\right)$ .

2. $G\circlearrowleft G$ by conjugation, $\mathrm{orb}(x)=\{g\bullet x:g\in G\}=\{gx{g}^{-1}:g\in G\}=$ the conjugacy class of $x$ (recall from the class equation).

3. $({\mathbb{Z}}_2,+)$ acts on the circle $\mathbb{S}1=◯$ If $x$ is some point on the circle, then $0\bullet x=x$ and $1\bullet x=$ the antipodal point on the circle. $\mathrm{orb}(x)=\{0\bullet x,1\bullet x\}=\{x,-x\}$

See

• Note - Orbits and Stabilizers