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The category of commutative affine algebraic group schemes over any field is commutativeabelian. More generally, the standard isomorphism theorems in abstract group theory hold in the category of affine algebraic group schemes over a field. See, for example, Chapt 1 of my online notes. Over other bases, this need not be true. If you are foolish enough to work with reduced algebraic group schemes (Borel, Humphreys, Springer, et al) then you run into all sorts or of problems.

Added: In 1962, Cartier gave a conference talk in which he noted that the standard isomorphism theorems etc. fail with the usual definition of algebraic groups (no nilpotents), but then observed that the "preceding difficulties vanish" when one works with schemes. As far as I know, this is the first statement in print. Of course, it is all covered in the 1963-64 Grothendieck-Demazure seminar (SGA3).

Over arbitrary bases, you can still form kernels, but quotients are a problem because the subgroup need not be flat. As noted, there are also problems when you drop the condition "affine and finite type".

When you don't allow nilpotents (i.e., when you work in the category of reduced group schemes), the Frobenius map $\mathbb{A}^1\to \mathbb{A}^1$ (this is a homomorphism when $\mathbb{A}^1$ is regarded as the additive group scheme) is mono and epi but not an isomorphism (no inverse). In the full category of affine algebraic group schemes, it has a kernel, namely, the finite group scheme $\alpha_p$, so there is no problem.

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The category of commutative affine algebraic group schemes over any field is commutative. More generally, the standard isomorphism theorems in abstract group theory hold in the category of affine algebraic group schemes over a field. See, for example, Chapt 1 of my online notes. Over other bases, this need not be true. If you are foolish enough to work with reduced algebraic group schemes (Borel, Humphreys, Springer, et al) then you run into all sorts or problems.