Probabilistically nilpotent groups of class two.

For G a finite group, let d 2 (G) denote the proportion of triples (x , y , z) ∈ G 3 such that [ x , y , z ] = 1 . We determine the structure of finite groups G such that d 2 (G) is bounded away from zero: if d 2 (G) ≥ ϵ > 0 , G has a class-4 nilpotent normal subgroup H such that [G : H] and | γ 4 (H) | are both bounded in terms of ϵ . We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite. [ABSTRACT FROM AUTHOR]