MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer
Thanks. Thats exactly what I was looking for. – Amritanshu Prasad Oct 26 '10 at 8:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.