Parabolic Subgroup
Parabolic subgroup may refer to: * a parabolic subgroup of a reflection group * a subgroup of an algebraic group that contains a Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgr ...

Parabolic Subgroup Of A Reflection Group
In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set \, which is generated by the set S = \ of adjacent transpositions, a subgroup is a ''standard parabolic subgroup'' if it is generated by a subset of . The ''parabolic subgroups'' of the symmetric group include the standard parabolic subgroups as well as all of their Conjugacy_class#Conjugacy_of_subgroups_and_general_subsets, conjugates. The symmetric group (generated by the adjacent transpositions) belongs to a larger family of reflection groups called ''Coxeter groups'', each of which comes with a special generating set . In this larger family, the standard parabolic subgroups are defined to be the subgroups generated by a subset of the special generating set. Separately, the symmetric group belongs to a larger family of reflection groups called ''complex reflection groups'', which are defined in terms of their action on ce ...
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