# 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 |

#### Comments

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

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

#### References

[a1] | I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) MR0093654 Zbl 0083.03301 |

[a2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039 |

[a3] | A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201 |

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Lie–Kolchin theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lie%E2%80%93Kolchin_theorem&oldid=22742