Exercise 5

Let $G$ be a group that acts on a non-empty set $X$. Prove that

1. every orbit is non-empty;

2. every element in $X$ is in some orbit; and

3. if $\mathrm{Orb}_{G}(x_{1})\cap\mathrm{Orb}_{G}(x_{2})\neq\varnothing$ where $x_{1},x_{2}\in X$, then $\mathrm{Orb}_{G}(x_{1})=\mathrm{Orb}_{G}(x_{2})$.