Probabilistically nilpotent groups of class two

Abstract

AbstractFor G a finite group, let $$d_2(G)$$ d 2 ( G ) denote the proportion of triples $$(x, y, z) \in G^3$$ ( x , y , z ) ∈ G 3 such that $$[x, y, z] = 1$$ [ x , y , z ] = 1 . We determine the structure of finite groups G such that $$d_2(G)$$ d 2 ( G ) is bounded away from zero: if $$d_2(G) \ge \epsilon > 0$$ d 2 ( G ) ≥ ϵ > 0 , G has a class-4 nilpotent normal subgroup H such that [G : H] and $$|\gamma _4(H)|$$ | γ 4 ( H ) | are both bounded in terms of $$\epsilon $$ 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.