# Definition(s) > [!NOTE] Definition (Fixed Points for Group Action) > The set of points of $X$ fixed by a [[Groups|group action]] $*:G\times X\to X$ are called the group's set of fixed points, defined by > $\text{Fix}(G) = \{x \in X: \text{for all } g \in G, \; g*x=x\}.$ > > > [!NOTE] Definition 2 (Characterisation by Orbits) > Then $\text{Fix}(G) = \{ x\in X: \text{Orb}_{G}(x)=\{ x \} \}$ where $\text{Orb}_{G}(x)$ denotes the [[Orbit under Group Action|orbit]] of $x$ under $*$. This is clearly equivalent to definition 1. > [!NOTE] Definition 2 (Characterisation by Stabiliser) > Then $\text{Fix}(G) = \{ x\in X: \text{Stab}_{G}(x)=\{ G \} \}$ where $\text{Stab}_{G}(x)$ denotes the [[Stabilizer under Group Action|stabiliser]] of $x$ under $*$. This is clearly equivalent to definition 1. # Properties Let $C_{p}$ be a set acting on $X$. Then $\#\text{Fix}(C_{p})=\#X$. We can prove this as follows. By [[Orbit-Stabilizer Theorem]], for all $x\in X$, $\text{Orb}_{G}(x)$ divides $\#C_{p}=p$ and so $\#\text{Orb}_{G}(x)$ equals either $1$ or $p$. Let $n$ be the number of distinct orbits with $1$ element and $m$ be the number of distinct orbits with $p$ elements. Since [[Orbits Under Group Action are Equivalence Classes|these orbits form a partition]] of $X$, we have $\#X=n+mp\equiv n\pmod{p}$. This completes the proof since, by definition, $n= \#\{ x\in X : \text{Orb}(x)=x \} = \# \text{Fix}(C_{p}).$