# Parabolic subgroup

From Groupprops

*The following property can be evaluated for a closed subgroup of an algebraic group*

## Definition

### Symbol-free definition

A closed subgroup of an algebraic group is termed a **parabolic subgroup** if it satisfies the following equivalent conditions:

- It contains a Borel subgroup of the whole group
- The quotient space , with the induced variety structure, is a complete variety.