Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)


Why is $G = G\_1\sqcup G\_2$, where $G\_1 \to U$ is finite and $G\_2\to U$ lies over Spec $K$?



Decomposing quasi-finite separated group schemes
[At OP's suggestion, I am moving my comment to an answer. In the original question $U$ was the spectrum of a Henselian DVR $R$ with $K = \mathrm{Frac}(R)$.]


Define $G\_1$ to be the union of those connected components of $G$ whose images cover $U$. Then use [EGA $\text{IV}\_4$, 18.5.11 c)] to verify that $G\_1$ is a clopen $U$-finite subgroup scheme of $G$.


