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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?

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up vote 8 down vote accepted

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

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Thanks. Thats exactly what I was looking for. –  Amritanshu Prasad Oct 26 '10 at 8:43
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