kernel of G(Z/p^2 Z)->G(Z/pZ) is the lie algebra of G over Z/pZ?

Question

Let $G$ be an affine algebraic group defined over $\mathbf Z$. The kernel of the natural homomorphism $G(\mathbf Z/p^2\mathbf Z)\to G(\mathbf Z/p\mathbf Z)$, if abelian, is a group which comes along with the conjugation action of $G(\mathbf Z/p\mathbf Z)$.

In the case where $G$ is a classical group, this kernel is isomorphic (as a set with $G(\mathbf Z/p\mathbf Z)$-action) to the Lie algebra $\mathfrak g(\mathbf Z/p\mathbf Z)$ of $G(\mathbf Z/p\mathbf Z)$ (which comes with the adjoint action). It seems that this should be the case in general.

Does anyone know of a reference for this kind of thing?

Answer 1

Take a look at Waterhouse's book - Introduction to affine group schemes. I think Theorem 12.2 is what you're looking for.