Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it generates a compact subgroup of $G(K)$ (equivalently, if $g$ is contained in a compact open subgroup of $G(K)$). Let $G(K)^o$ be the (normal and open) subgroup of $G(K)$ generated by the compact elements. Let $Z$ be the center of $G(K)$. I am looking for proofs or references for the following facts:

1. $G(K)/G(K)^o$ is a finitely generated free abelian group.
2. The image of $Z$ in $G(K)/G(K)^o$ is of finite index.
3. $Z \cap G(K)^o$ is a maximal compact subgroup of $Z$.

Answers handling any subset of these claims are very welcome.

Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact subgroup of $G(K)$ (equivalently, if $g$ is contained in a compact open subgroup of $G(K)$). Let $G(K)^o$ be the (normal and open) subgroup of $G(K)$ generated by the compact elements. Let $Z$ be the center of $G(K)$. I am looking for proofs or references for the following facts:

1. $G(K)/G(K)^o$ is a finitely generated free abelian group.
2. The image of $Z$ in $G(K)/G(K)^o$ is of finite index.
3. $Z \cap G(K)^o$ is a maximal compact subgroup of $Z$.

Answers handling any subset of these claims are very welcome.