Parabolic subgroup

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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.