Unipotent normal subgroups of algebraic groups

Abstract

Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical ${\mathrm{Rad}}_u(G)$ of $G$. We investigate some properties of ${\mathrm{Rad}}_u(G)$, and study those $G$ for which ${\mathrm{Rad}}_u(G)$ is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine $k$-groups.