In order to understand the structure of reductive groups, we will first look at some "base cases" of groups that are quite small. These are the groups of so-called semisimple rank 1, which by definition are the algebraic groups $ G$ such that $ G/R(G)$ has rank 1, where $ R(G)$ is the connected component of the unique largest normal solvable subgroup. In this post, we shall see a detailed proof of a theorem that gives several different characterisations of these groups.

As usual, we consider algebraic groups over an algebraically closed field $ k$. The proof we follow will be Theorem 25.3 in Humphrey's book "Linear Algebraic Groups". The reason I will go through it here is because in the book, I found the proof a bit terse and in a few points the proof relies on exercises, so it should be instructive to write down a more self-contained proof in my own words.

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