The derived subgroup, the derived series and solvable algebraic groups

Let $k$ be a field and let $G$ be an algebraic group over $k$, that is, a group scheme of finite type over $k$ (Group schemes of finite type over a field), with multiplication $m$, inverse $i$ and identity section $e$.

The commutator morphism is $c:G{\times }_{k}G\to G,c(g,h)=gh{g}^{-1}{h}^{-1}\mathrm{.}$ It is a $k$-morphism: the map $(g,h)\mapsto (g,h,g^{-1},h^{-1})$ is built from $i$ and the universal property of the fibre product (Fibre product of schemes), and $c$ is obtained from it by iterated multiplication. For every $k$-algebra $R$, the induced map $G(R)\times G(R)\to G(R)$ is the group commutator.

The derived subgroup $DG$ is the smallest closed subgroup scheme $H\subseteq G$ (Morphisms and closed subgroup schemes of group schemes) through which $c$ factors, that is, such that $c$ equals the composite of a morphism $G{\times }_{k}G\to H$ with the inclusion $H\hookrightarrow G$. Equivalently, $DG$ is the smallest closed subgroup scheme of $G$ containing the scheme-theoretic image of $c$ (Scheme-theoretic image, Scheme-theoretic image of a quasi-compact morphism). It exists: if $G$ is affine, closed subgroup schemes correspond to Hopf ideals of $O(G)$ and $c$ corresponds to a $k$-algebra map ${\mathrm{\Delta }}_c:O(G)\to O(G){\otimes }_{k}O(G)$, and the Hopf ideals contained in $\ker {\mathrm{\Delta }}_c$ are closed under sums, so their largest element cuts out $DG$; in general the closed subgroup schemes through which $c$ factors are closed under schematic intersection, their intersection is computed by the ideal sheaf generated by the defining ideal sheaves, and $G$ is Noetherian because it is of finite type over a field.

The derived series of $G$ is the sequence $D^{0}G=G,D^{i+1}G=D(D^{i}G)(i\ge 0),$ so each $D^{i}G$ is a closed subgroup scheme of $G$ for $i\ge 1$. The group $G$ is solvable if $D^{n}G=1$ for some $n\ge 0$, where $1$ denotes the trivial subgroup scheme, the image of the identity section $e$.

A closed subgroup scheme $N\subseteq G$ is normal if it is stable under the conjugation action of $G$ on itself: the morphism $G{\times }_{k}N\to G$, $(g,n)\mapsto gn{g}^{-1}$, factors through the inclusion $N\hookrightarrow G$. For $k$-algebras $R$ this says exactly that $N(R)$ is a normal subgroup of $G(R)$. This is the sense of normality used throughout this page.