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In this very short note I want to summarise some basic facts about algebraic groups which put clear some issues which have confused me a several times in the last few months. I will not give any proofs, and am just picking out remarks from Brian Conrad’s paper `A Modern Proof of Chevalley’s Theorem on Algebraic Groups.’

An algebraic group $G$ over $k$ is what you think it is: a connected smooth group scheme over $k$: such a $G$ is automatically a geometrically integral variety.

What obvious special classes of $G$ are there? We could have $G$ affine. In fact, any such group is a closed algebraic subgroup of a matrix group $(GL_n)_k$, so we call the affine algebraic groups linear algebraic groups.

Alternatively, one could imagine a different class: all those $G$ which are proper over $k$, and by definition these are the abelian varieties. That these are actually abelian as groups on geometric points follows from the rigidity lemma for proper schemes (as in Mumford’s book).

Suppose now that $G$ is an arbitrary algebraic group over $k$. Chevalley’s theorem tells us that provided $k$ is perfect, we can find a unique normal algebraic subgroup $H$ of $G$ with $H$ affine  and $G/H$ proper. In other words, there is a unique (up to canonical isomorphism) exact sequence:

$0 \rightarrow H \rightarrow G \rightarrow A \rightarrow 0$

with $H$ a linear algebraic group, and $A$ an abelian variety.