View source for Lie-Kolchin theorem

A solvable subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587101.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587102.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587103.png" /> is a finite-dimensional vector space over an algebraically closed field) has a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587104.png" /> of index at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587106.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587107.png" />, such that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587108.png" /> there is a [[Flag|flag]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l0587109.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871010.png" />. In other words, there is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871011.png" /> in which the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871012.png" /> are written as triangular matrices. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871013.png" /> is a connected closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871014.png" /> in the Zariski topology, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871015.png" />; in this case the Lie–Kolchin theorem is a generalization of Lie's theorem, which was proved by S. Lie for complex connected (in the Euclidean topology) solvable Lie groups (see [[Lie group, solvable|Lie group, solvable]]; [[Lie theorem|Lie theorem]]). This assertion can also be considered as a special case of Borel's fixed-point theorem (cf. [[Borel fixed-point theorem|Borel fixed-point theorem]]).

The following analogue of the Lie–Kolchin theorem is true for an arbitrary field: A solvable group of matrices contains a normal subgroup of finite index whose commutator subgroup is nilpotent.

The Lie–Kolchin theorem was proved by E.R. Kolchin [[#References|[1]]] (for connected groups) and A.I. Mal'tsev [[#References|[2]]] (in the general formulation). It is also sometimes called the Kolchin–Mal'tsev theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.R. Kolchin,   "Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations"  ''Ann. of Math. (2)'' , '''49'''  (1948)  pp. 1–42</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. [A.I. Mal'tsev] Mal'Avcev,   "On certain classes of infinite soluble groups"  ''Transl. Amer. Math. Soc. (2)'' , '''2'''  (1956)  pp. 1–21  ''Mat. Sb.'' , '''28'''  (1951)  pp. 567–588</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====
In Western literature the Lie–Kolchin theorem usually designates the more restricted version about connected closed subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058710/l05871016.png" />.

For the role of the Lie–Kolchin theorem in the Galois theory for ordinary linear differential equations see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kaplansky,   "An introduction to differential algebra" , Hermann  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Humphreys,   "Linear algebraic groups" , Springer  (1975)  pp. Sect. 35.1</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)  pp. 283ff</TD></TR></table>