Editing Linear algebraic group (section)

===Basic notions===
For a linear algebraic group ''G'' over a field ''k'', the [[identity component]] ''G''<sup>o</sup> (the [[connected component (topology)|connected component]] containing the point 1) is a [[normal subgroup]] of finite [[index of a subgroup|index]]. So there is a [[group extension]]
:<math>1 \to G^\circ \to G \to F \to 1, </math>
where ''F'' is a finite algebraic group. (For ''k'' algebraically closed, ''F'' can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.

Various notions from [[group theory|abstract group theory]] can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be [[abelian group|commutative]], [[nilpotent group|nilpotent]], or [[solvable group|solvable]], by analogy with the definitions in abstract group theory. For example, a linear algebraic group is '''solvable''' if it has a [[composition series]] of linear algebraic subgroups such that the quotient groups are commutative. Also, the [[normalizer]], the [[center of a group|center]], and the [[centralizer]] of a closed subgroup ''H'' of a linear algebraic group ''G'' are naturally viewed as closed subgroup schemes of ''G''. If they are smooth over ''k'', then they are linear algebraic groups as defined above.

One may ask to what extent the properties of a connected linear algebraic group ''G'' over a field ''k'' are determined by the abstract group ''G''(''k''). A useful result in this direction is that if the field ''k'' is [[perfect field|perfect]] (for example, of characteristic zero), ''or'' if ''G'' is reductive (as defined below), then ''G'' is [[unirational]] over ''k''. Therefore, if in addition ''k'' is infinite, the group ''G''(''k'') is [[Zariski dense]] in ''G''.<ref>Borel (1991), Theorem 18.2 and Corollary 18.4.</ref> For example, under the assumptions mentioned, ''G'' is commutative, nilpotent, or solvable if and only if ''G''(''k'') has the corresponding property.

The assumption of connectedness cannot be omitted in these results. For example, let ''G'' be the group μ<sub>''3''</sub> ⊂ ''GL''(1) of cube roots of unity over the [[rational number]]s '''Q'''. Then ''G'' is a linear algebraic group over '''Q''' for which ''G''('''Q''') = 1 is not Zariski dense in ''G'', because <math>G(\overline {\mathbf Q})</math> is a group of order 3.

Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a [[rational variety]].<ref>Borel (1991), Remark 14.14.</ref>