Editing Perfect group (section)

== Ore's conjecture ==
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 [[conjecture]]d that this was so for all the finite non-abelian simple groups. Ore's conjecture was finally [[mathematical proof|proven]] in 2008. The proof relies on the [[classification of finite simple groups|classification theorem]].<ref>{{cite journal|authorlink1=Martin Liebeck |last1=Liebeck |first1=Martin |last2=O'Brien |first2=E.A. |last3=Shalev |first3=Aner |authorlink3=Aner Shalev |last4=Tiep |first4=Pham Huu |authorlink4=Pham Huu Tiep |title=The Ore conjecture|url=https://www.math.auckland.ac.nz/~obrien/research/ore.pdf|journal=[[Journal of the European Mathematical Society ]] |volume=12|year=2010|issue=4 |pages=939–1008|doi=10.4171/JEMS/220 |doi-access=free}}</ref>