---
tag: DNE3N4DCC5
kind: statement
role: theorem
name: Tits's simplicity theorem
sources:
  - wikipedia: en:1772133@r1346225273
    locator: (B, N) pair, section Properties > Tits's simplicity theorem
    note: adopted
    mention: wikipedia://en/1772133#8BPD
    sections: [Statement]
inventory: [1YXNN0CNS5]
---

## Statement

BN-pairs can be used to prove that many groups of Lie type are [[simple|0C093D1T4M]] modulo their [[centers|0DZ93493VN]]. More precisely, if *G* has a *BN*-pair such that *B* is a [[solvable group|0FXJ7YFMMX]], the intersection of all conjugates of *B* is trivial, and the set of generators of *W* cannot be decomposed into two [[non-empty|0ECVK59Y03]] commuting sets, then *G* is simple whenever it is a perfect group. In practice all of these conditions except for *G* being perfect are easy to check. Checking that *G* is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showing that a group is perfect is usually far easier than showing it is simple.