I am reading Tapp's intro to matrix groups for undergraduates. On page 46 he states the following theorem:
For $X\subseteq \mathbb R^2$ if $Symm(X)$ is finite then it is isomorphic to $D_m$ or $\mathbb Z_m$ for some $m$.
Following it he writes:
The proof involves two steps. First, when $Symm(X)$ is finite its elements must share a common fixed point.
But he does not elaborate and it is not obvious to me. Why is this clear?