# Lie-Kolchin theorem

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A solvable subgroup $G$ of the group $\mathop{\rm GL} ( V)$( where $V$ is a finite-dimensional vector space over an algebraically closed field) has a normal subgroup $G _ {1}$ of index at most $\rho$, where $\rho$ depends only on $\mathop{\rm dim} V$, such that in $V$ there is a flag $F = \{ V _ {i} \}$ that is invariant with respect to $G _ {1}$. In other words, there is a basis in $V$ in which the elements of $G _ {1}$ are written as triangular matrices. If $G$ is a connected closed subgroup of $\mathop{\rm GL} ( V)$ in the Zariski topology, then $G _ {1} = G$; 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 theorem). This assertion can also be considered as a special case of Borel's fixed-point theorem (cf. 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  (for connected groups) and A.I. Mal'tsev  (in the general formulation). It is also sometimes called the Kolchin–Mal'tsev theorem.

How to Cite This Entry:
Lie–Kolchin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie%E2%80%93Kolchin_theorem&oldid=22742