Editing Linear algebraic group (section)

==Unipotent groups==
Let ''U''<sub>''n''</sub> be the group of upper-triangular matrices in ''GL''(''n'') with diagonal entries equal to 1, over a field ''k''. A group scheme over a field ''k'' (for example, a linear algebraic group) is called '''unipotent''' if it is isomorphic to a closed subgroup scheme of ''U''<sub>''n''</sub> for some ''n''. It is straightforward to check that the group ''U''<sub>''n''</sub> is nilpotent. As a result, every unipotent group scheme is nilpotent.

A linear algebraic group ''G'' over a field ''k'' is unipotent if and only if every element of <math>G(\overline{k})</math> is unipotent.<ref>Milne (2017), Corollary 14.12.</ref>

The group ''B''<sub>''n''</sub> of upper-triangular matrices in ''GL''(''n'') is a [[semidirect product]]
:<math>B_n = T_n \ltimes U_n,</math>
where ''T''<sub>''n''</sub> is the diagonal torus (''G''<sub>''m''</sub>)<sup>''n''</sup>. More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group, ''T'' ⋉ ''U''.<ref>Borel (1991), Theorem 10.6.</ref>

A smooth connected unipotent group over a perfect field ''k'' (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group ''G''<sub>''a''</sub>.<ref>Borel (1991), Theorem 15.4(iii).</ref>