---
tag: 9P77XM5E8C
kind: concept
name: Parabolic subgroup of a reflection group
sources:
  - wikipedia: en:75737969@r1364006660
    locator: Parabolic subgroup of a reflection group
    note: adopted
    mention: wikipedia://en/75737969
    sections: [Summary]
same_as:
  wikidata: Q124328767
  wikipedia: Parabolic subgroup of a reflection group
inventory: [9247G9N3AH]
---

## Summary

In the mathematical theory of reflection groups, the **parabolic subgroups** are a special kind of [[subgroup|0X1JXT47GA]]. In the [[symmetric group|0515A6D2Y1]] of permutations of the set $\{1, 2, \ldots, n\}$, which is generated by the set $S = \{(1\ 2), (2\ 3), \ldots, (n - 1\ n)\}$ of adjacent transpositions, a subgroup is a *standard parabolic subgroup* if it is generated by a subset of $S$; equivalently, they are the groups of permutations that come from partitioning the set $\{1, 2, \ldots, n\}$ into parts $\{1, 2, \ldots, a_1\}$, $\{a_1 + 1, a_1 + 2, \ldots, a_1 + a_2\}$, etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself. The *parabolic subgroups* of the symmetric group include the standard parabolic subgroups as well as all of their [[conjugates|0DEBXAJ5SA]].