Parabolic subgroup of a reflection group

In the mathematical theory of reflection groups, a parabolic subgroup is a special kind of subgroup. The precise definition of which subgroups are parabolic depends on context—for example, whether one is discussing general Coxeter groups or complex reflection groups—but in all cases the collection of parabolic subgroups exhibits important good behaviors, and the different definitions essentially coincide in the case of finite real reflection groups. For example, the parabolic subgroups of a reflection group have a natural indexing set and form a lattice when ordered by inclusion. Parabolic subgroups arise in the theory of algebraic groups, through their connection with Weyl groups.