Can any arbitrary group act on any arbitrary set?

Question

Usually the examples for group actions involve a permutation group $(S_n)$ of a set (with $n$ elements) acting on that set. But what about $S_n$ acting on a set that has more or less elements than $n$ ? How does that make sense ?

Answer 1

Let $G$ be a group and $X$ be a set.

---

All that is required for a map $\cdot:G\times X\to X$ to be an action is:

• for all $g,h\in G, x\in X$, $$g\cdot(h\cdot x)=(gh)\cdot x,$$
• for all $x\in X$, $$e\cdot x=x.$$

---

Consider the map defined by $k\cdot y=y$ for all $k\in G, y\in X$. Then, for $g,h\in G, x\in X$, we have

$$\begin{align} g\cdot(h\cdot x)&=g\cdot x\\ &=x\\ &=(gh)\cdot x \end{align}$$

and trivially

$$e\cdot x=x.$$

Hence $G\curvearrowright X$.