You need to know: Group, finite group, identity element e of a group, notation for the inverse of group element g, subgroup.
Background: A subgroup S of group G is called normal, if for any and . A group G is called simple if it does not have any normal subgroup, except of itself and , where is the set consisting on identity element only. A group G is called abelian if for every and , and non-abelian otherwise. An element is called a commutator if for some and .
The Theorem: On 18th June 2009, Martin Liebeck, Eamonn O’Brien, Aner Shalev, and Pham Huu Tiep submitted to the Journal of the European Mathematical Society a paper in which they proved that every element of every finite non-abelian simple group is a commutator.
Short context: In any abelian group, , hence only identity element is a commutator. In general, the size of set of commutators can be viewed as an indicator “how far” the group is from being abelian. In 1951, Ore made a conjecture that in every finite non-abelian simple group the set of commutators is the whole group, which makes such groups as “far” from being abelian as they possibly can. Despite substantial efforts, the conjecture remained open for 58 years. The Theorem proves this conjecture.
Theorems of the 21st century 
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