Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

Abstract

For a finite smooth algebraic group $F$ over a field $k$ and a smooth algebraic group $\bar G$ over the separable closure of $k$, we define the notion of $F$-kernel in $\bar G$ and we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of $F$ by an arbitrary smooth $k$-group $G$. We show in particular that any such extension comes from an extension of finite $k$-groups when $k$ is perfect and we give explicit bounds on the order of these finite groups when $G$ is linear. We prove moreover some finiteness results on these sets.