Parabolic subgroup

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

Definition

Symbol-free definition

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

1. It contains a Borel subgroup of the whole group
2. The quotient space $G/P$, with the induced variety structure, is a complete variety.