# Maximal unipotent subgroup

From Groupprops

## Contents

## Definition

A **maximal unipotent subgroup** of a linear algebraic group is a subgroup satisfying the following equivalent conditions:

- It is maximal among connected unipotent closed subgroups of the whole group. In other words, it is a closed connected subgroup in which all elements are unipotent but that is not contained in any bigger closed connected subgroup in which all elements are unipotent.

## Facts

When the base field is algebraically closed all the maximal unipotent subgroups form a single conjugacy class. In other words, upto conjugacy, we can talk of **the** maximal unipotent subgroup.

## Particular cases

Group | Maximal unipotent subgroup |
---|---|

general linear group | unitriangular matrix group |

special linear group | unitriangular matrix group |

symplectic group | maximal unipotent subgroup of symplectic group |

## Analogues

The analogue of maximal unipotent in finite group theory is the notion of -Sylow subgroup. In fact, in the case of an algebraic group over a finite field (which may better be thought of as the finite field-points of the algebraic group over its algebraic closure), the maximal unipotent subgroup *is* the -Sylow subgroup.