A Primer on Group Actions

Definition: Group Actions

A group $G$ is said to act on a set $X$ if we have a map $(G,X)\to X$ (which we write as $g\cdot x$) that satisfies:

• $1_g\cdot x=x$ for all $x\in X$
• $g_1\cdot (g_2\cdot x)=(g_1{g}_2)\cdot x$ for all $g_1,g_2\in G$ and $x\in X$

Example Fest §

1. $\mathbb{Z}/n\mathbb{Z}$ acting on $\mathbb{C}$ by $k\cdot z=\exp \left(\frac{2\pi ik}{n}\right)$ (i.e., rotating $z$ by $\frac{2\pi k}{n}$)
2. $\mathbb{Z}/n\mathbb{Z}$ acting similarly on ${\mathbb{R}}^2$ by sending $k\cdot v$ to the rotation matrix by $\frac{2\pi k}{n}$.