---
tag: XFAXZ3ER55
kind: statement
role: proposition
name: Ore's conjecture
sources:
  - wikipedia: en:585271@r1284492210
    locator: Perfect group, section Ore's conjecture
    note: adopted
    mention: wikipedia://en/585271#FYZB
    sections: [Statement]
inventory: [WD85BAEN1J]
---

## Statement

As the commutator subgroup is *generated* by commutators, a perfect group may contain elements that are products of commutators but not themselves commutators. Øystein Ore showed in 1951 that the alternating groups on five or more elements contained only commutators, and conjectured that this was so for all the finite non-abelian simple groups. Ore's conjecture was finally [[proven|0B94TKFQJX]] in 2008. The proof relies on the [[classification theorem|06J6P03MG7]].