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  "body": "\n## Summary\n\nIn 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]].\n",
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