# A question on periodic points and recurrent points

I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a topological group and $G_x$ is the isotropy group of $x$. And a point $x$ is recurrent if for each neightborhood $U$ of $x$ and each compact $K\subset G$ there is $g$ in the complement of $K$ such that $gx\in U$. According to the book, periodic points are recurrent. I do not see why this holds obviously.

I have also find some other definitons of periodic points and recurrent points by using the term "syndetic". And I also do not kown how to prove that the definitions are equivalent?

Thank you!