Maximal unipotent subgroup

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Definition

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

  1. 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 p-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 p-Sylow subgroup.