Normal closed subgroup

Prove that an algebraic G is triangulable (9, Ex. 6) iff it has a unipotent normal closed subgroup U with G/U diagonalizable. [If G acts on V, then U acts trivially on a nonzero subspace V0. The map G - Aut(V0) factors through G/U, which will have an eigenvector.]... 

Let / be an ideal in a Hopf algebra A. Work out the conditions necessary for A/l to represent a closed subgroup which is normal. 

Consider connected closed subgroups H of G which are normal and solvable. If Ht and H2 are such, so is (the closure of) H H, since the dimensions cannot increase forever, there is actually a largest such subgroup. We denote it by R and call it the radical of G. By (10.3), the unipotent elements in R form a normal subgroup U, the unipotent radical. We call G semisimple if R is trivial, reductive if U is trivial. The theorem then (for char(fc) = 0) is that all representations are sums of irreducibles iff G is reductive. It is not hard to see this condition implies G reductive (cf. Ex. 20) the converse is the hard part. We of course know the result for R, since by (10.3) it is a torus we also know that this R is central (7.7), which implies that the R-eigenspaces in a representation are G-invariant. The heart of the result then is the semisimple case. This can for instance be deduced from the corresponding result on Lie algebras. 

Theorem. Let G be an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

Corollary. Let S be any solvable matrix group over an algebraically closed field. Then S has a normal subgroup of finite index which can be put in triangular form. 

Theorem. Let G be an affine group scheme over a field. Let N be a closed normal subgroup. Then there is a quotient map G - H with kernel precisely N. 

Corollary. There is a one-to-one correspondence between closed normal subgroups and quotients. 

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