Definition of Unipotent in Positive Characteristic Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$.  Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all elements $g\in G$ such that $g^{p^t}=1$ for some $t\in\mathbb{N}$?  If not, what is $\mathcal{U}(G)$?  If so, what are equivalent definitions of $\mathcal{U}(G)$ and why are they equivalent?
I'm just trying to grow in my understanding of the definition of unipotence.  Thanks for your help!
 A: The set of unipotent elements are indeed the elements whose orders are some power of the characteristic of the field.
To see this, let $g\in G$ and embed $G$ in some general linear group $H$ over the field $k$. Let $p$ be the characteristic of $k$.
Since the order of the image of $g$ in $H$ is a power of $p$ if and only if the order of $g$ is a power of $p$, we just need to show that a matrix over a field of characteristic $p$ is unipotent if and only if it has order a power of $p$.
Let $A$ be a unipotent matrix, so $(A-I)^m = 0$ for some $m$. Let $n$ be given such that $p^n\geq m$. Then we have $0 = (A - I)^{p^n} = A^{p^n} - I$ so $A$ has order a power of $p$.
On the other hand, if we assume that $A^{p^n} = I$ for some $n$ then we have $(A - I)^{p^n} = A^{p^n} - I = 0$ so $A$ is unipotent.
A: More "intrinsically", although you still need an embedding, is to embed $G \to GL(k[G])$. (We assume $k$ is algebraically closed). Here $k[G]$ is the ring of regular functions on $G$, and $G$ acts on $k[G]$ by
$$(g\cdot f)(x) = f(g^{-1}x)$$
Of course, $k[G]$ is an infinite dimension vector space, but you can show that it is the direct limit of finite dimensional $G$-stable subspaces. An unipotent element in $GL(k[G])$ is then defined to be an element unipotent on any restriction to the finite dimensional $G$-stable pieces. 
But then for this definition to be used, you would need to check that under a morphism of algebraic groups, unipotent element gets sent to a unipotent one. Also for $GL_n$, unipotence in the usual sense is the same as this sense.
