# Lie-Kolchin theorem

(Redirected from Lie–Kolchin theorem)

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 [1] (for connected groups) and A.I. Mal'tsev [2] (in the general formulation). It is also sometimes called the Kolchin–Mal'tsev theorem.

#### References

 [1] 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 [2] 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 [3] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) MR0551207 Zbl 0549.20001

In Western literature the Lie–Kolchin theorem usually designates the more restricted version about connected closed subgroups of $\mathop{\rm GL} ( V)$.