Prove that the affine transformation $f$ has at least one fixed point,

Question

If $f:X \rightarrow\ X$ is an bijective affine transformation of finite order ( an affine automorphism of finite order) in the group of affine automorphisms of $X$, prove that $f$ has at least one fixed point. I supposed that for every $x \in X$, $f(x)\neq x$, so $F_{f}=\{M\in X:f(M)=M\}$ has the property $F_{f}=\emptyset$. $F_{f}=ker(f`-id_{\vec{x}})$, so because it is null, $f`(x) \neq id_{\vec{x}}$, but I don`t know how to prove what I am required. Any help, please?

Answer 1

Let $p$ be the order of $f$ and $x\in X$, write $y={1\over p}(x+f(x)+..+f^{p-1}(x))$, $f(y)=y$.